Spectra of Groups

The aim of this paper is to investigate the behaviour of prime and semiprime subgroups of groups, and their relation with the existence of abelian normal subgroups. In particular, we study the set Spec(G) of all prime subgroups of a group G endowed with the Zariski topology and, among other examples, we construct an infinite group whose proper normal subgroups are prime and form a descending chain of type ω + 1.


Introduction
It was shown in [6] that it is possible to associate a Zariski spectrum with any complete multiplicative lattice.In this paper, we associate with any group G the Zariski spectrum Spec(G) of the multiplicative lattice N (G) of all normal subgroups of G.The multiplication in this lattice is given by the commutator operation, i.e., the product of two normal subgroups A and B of G is the normal subgroup [A, B].The points of the spectrum Spec(G) of G are the prime subgroups of G, i.e. the prime elements of the multiplicative lattice N (G) (see next section for the precise definition).Similarly, a normal subgroup of G is called semiprime if it is a semiprime element of N (G).
In this way, one gets a theory very similar to that of the Zariski spectrum of a commutative ring without identity or to that of the Zariski spectrum of a noncommutative ring.It turns out, for instance, that a subgroup N of a group G is semiprime if and only if it is the intersection of a (possibly empty) family of prime subgroups of G. Thus the intersection of all prime subgroups coincides with the intersection of all semiprime subgroups of G, and it is proved to be the largest normal subgroup admitting an ascending G-invariant series with abelian factors.It follows that a group has an empty spectrum if and only if it is hyperabelian.
As in [6], the spectrum of a group G is endowed with the Zariski topology, in which the closed subsets are the sets V (N) = {P ∈ Spec(G) : N ≤ P }, N ∈ N (G).In such a way, Spec(G) becomes a sober topological space, which is not compact in general.The lattice (Spec(G)) of the open subsets of Spec(G) is isomorphic to the lattice of all semiprime subgroups of G (see [6]).Like in the case of commutative rings, decompositions of Spec(G) as a disjoint union of two clopen subsets are related to decompositions of G as a direct product of two subgroups.More generally, if N is any normal subgroup of G, we show that the study of Spec(G) reduces to that of Spec(G/N ) and Spec G (N ), where Spec G (N ) denotes the space of all G-prime subgroups of N (see Section 3 for details).
Several examples are given in order to illustrate the theory of prime and semiprime subgroups.In particular, we construct an infinite group all of whose proper normal subgroups are prime and form a descending chain of type ω + 1 (see Theorem 2.16), and for each ordinal α a group all of whose proper normal subgroups are prime and form an ascending chain of type α (see Theorem 3.10).
Finally, in Section 4, we introduce analogues for groups to m-systems and n-systems of rings (see [12, pp.166-167]), and we note that these concepts are strictly related to prime and semiprime subgroups.In contrast to the case of rings, we prove that not all n-systems of a group can be obtained as a union of m-systems (see Theorem 4.5).
Most of our notation is standard.In particular Sym(n) and Alt(n) denote the symmetryc group and the alternating group of degree n, respectively; moreover, if X is any set, FAlt(X) is the finitary alternating group over X.

Prime and Semiprime Subgroups
Let G be a group.The modular lattice of all normal subgroups of G is denoted by N (G).We say that a proper normal subgroup P of G is prime if, for every A, B ∈ N (G), [A, B] ≤ P implies either A ≤ P or B ≤ P .Prime subgroups were first introduced by Schenkman [18] in the case of groups satisfying the maximal condition on normal subgroups (see also [11,19,20]).The set of all prime subgroups of G is called the spectrum of G and is denoted by Spec(G).Notice that in the definition of a prime subgroup it is enough to consider only normal subgroups A and B containing P , because [AP , BP ] ≤ P if and only if [A, B] ≤ P .Thus a maximal normal subgroup P of a group G is prime if and only if the simple group G/P is not abelian.
We say that a normal subgroup N of G is semiprime if, for every normal subgroup A of G, A = [A, A] ≤ N implies A ≤ N ; also in this definition it is possible to consider only normal subgroups containing N .Thus N is semiprime if and only if the factor group G/N has no abelian nontrivial normal subgroups; in particular, the whole group G is semiprime in itself, while a maximal normal subgroup is prime if and only if it is semiprime.
Of course, every prime subgroup is semiprime, but the direct product of two simple nonabelian groups shows that there exist proper semiprime subgroups that are not prime.
The following result provides a characterization of prime subgroups among semiprime subgroups.Here we say that a group G is uniform if the lattice N (G) is uniform, that is, G is nontrivial and A ∩ B = {1} for all nontrivial normal subgroups A and B of G (see [7], where such groups are introduced and studied).
Proof Suppose that N is prime, so in particular it is also semiprime.If A and B are normal Conversely, assume that N is semiprime and In the study of prime and semiprime subgroups of a group G, one immediately notes that the image of a prime subgroup under a group epimorphism need not be even semiprime.In fact, if α is the automorphism of the direct product acting on each factor as the transposition (12), then A 1 a is prime subgroup of G = α A, while A 1 A 2 /A 2 = A/A 2 is not even semiprime in G/A 2 .On the other hand, similarly to the case of commutative rings without identity, the preimages of (semi)prime subgroups behave better.

Lemma 2.2 Let
Proof We prove the statement when P is prime.Let X and Y be normal subgroups of G 1 such that [X, Y ] ≤ f −1 (P ).Then f (X) and f (Y ) are normal subgroups of G 2 and It is also easy to see that primality can be detected from the behaviour of normal closures of cyclic subgroups.

Lemma 2.3 Let N be a proper normal subgroup of a group G. (1) N is prime if and only if, for all
The following result and a direct application of Zorn's lemma yield that every prime subgroup of a group contains a minimal prime subgroup.In particular, every group with a nonempty spectrum admits minimal prime subgroups.

Lemma 2.4
Let G be a group and let L be a nonempty chain of prime subgroups of G. Then the intersection P of all members of L is prime.
Proof Let A and B be normal subgroups of G such that [A, B] ≤ P .If A is not contained in P , then there is L 0 ∈ L such that A ≤ L 0 , and hence B ≤ L for every L ∈ L with L ≤ L 0 .It follows that B ≤ P and so P is a prime subgroup of G.
The example of a direct product of two simple nonabelian groups proves that the intersection of two prime subgroups need not be prime, although it is semiprime.In fact, it easily follows from the definition that the intersection of any collection of semiprime subgroups is semiprime.In particular, if N is a normal subgroup of a group G, then there is a smallest semiprime subgroup of G containing N .We call this subgroup the semiprime closure B(G : N) of N in G (cf. the remark after Corollary 2.7).Of course, the semiprime closure of the trivial subgroup is the smallest semiprime subgroup of G.
As opposed to Lemma 2.4, the union of a chain of prime subgroups can be a proper non-prime subgroup.To see this, consider a set X of cardinality ℵ ω and let S be the normal subgroup of Sym(X) consisting of all permutations whose support has cardinality strictly smaller than ℵ ω .Let g be an element of order 2 in Sym(X) fixing no elements of X and put G = g S. For each nonnegative integer k, consider the normal subgroup S k of G consisting of all elements with a support of cardinality at most ℵ k .It is well known that the subgroups S k together with the finitary alternating and finitary symmetric groups are the only nontrivial normal subgroups of S, so that, in particular, the lattice N (G) is a chain.Since S k+1 /S k is simple nonabelian for each k ≥ 0, it follows that every S k is a prime subgroup of G, while S = k S k is not even semiprime.The same example shows that a group with a nonempty spectrum may have no maximal prime subgroups, even if it is not the union of its prime subgroups.
Our next aim is to describe the semiprime closure.To this purpose we use a definition that was implicitly introduced by Reinhold Baer in his famous characterization of hyperabelian groups (see [3], or also [17] Part 1, Theorem 2.15).
Let x be an element of a group G.We say that x is a Baer element if for all sequences x = x 0 , x 1 , . . ., x n , . . .and y 0 , y 1 , . . ., y n , . . . of elements of G such that there is a nonnegative integer k such that x n = 1 for each n ≥ k.This concept should be seen in relation to the so-called vanishing m-sequences of elements of a ring introduced by Levitizki in [14].
The set of all Baer elements of G will be denoted here by B(G).Baer's theorem states that G is hyperabelian if and only if B(G) = G.

Lemma 2.5 Let G be a group and let 1 = x ∈ B(G). Then x G contains an abelian nontrivial G-invariant subgroup.
Proof Assume by contradiction that the statement is false, so that in particular there is an element y 0 of G such that [x y 0 0 , x 0 ] = 1, where x 0 = x.Suppose that for some n we have defined x n and y n in such a way that [x G is not abelian.This clearly contradicts the fact that x is a Baer element.
then G-prime subgroups are just the prime ones and Spec G (G) = Spec(G).Notice that a G-prime subgroup of H need not be prime in G as shown by considering the direct product G = A × B × C of three simple non-abelian groups: here, we refer to the G/K-prime subgroups of H/K simply as G-prime subgroups; moreover, as usual, we assume that the intersection of an empty collection of subsets of a set S is the set S itself.

Theorem 2.6 Let H be a normal subgroup of a group G. Then B H (G) := H ∩ B(G) is the intersection of all G-prime subgroups of H .
Proof Let P be a G-prime subgroup of H and assume that x is an element of B H (G) \ P .Then the coset xP is a Baer element of G/P and so by Lemma 2.5we have that H/P contains an abelian nontrivial G-invariant subgroup, which is impossible since P is Gprime.Thus P ⊇ B H (G) and hence B H (G) is contained in the intersection of all G-prime subgroups of H .
. By Zorn's Lemma, there exists a G-invariant subgroup M of H which is maximal with respect to the condition x n / ∈ M for all n.Let A and B be G-invariant subgroups of H such that [A, B] ≤ M and assume by contradiction that neither A nor B is contained in M.Then, by the maximal choice of M, there are non-negative integers h and k such that x h ∈ AM and x k ∈ BM.It follows that x h+k ∈ AM ∩ BM.Finally, and this contradiction proves that M is G-prime in H . Therefore the intersection of all G-prime subgroups of H lies in B H (G), and the statement is proved.
The above result has the following obvious consequence, showing in particular that B(G) is a semiprime subgroup of G, although it is not always prime.

Corollary 2.7 If G is any group, then B(G) is the intersection of all prime subgroups of G.
If N is a normal subgroup of a group G, Corollary 2.7 provides an explanation for the notation B(G : N) we used for the semiprime closure of N in G.In fact, it follows from this result that B(G : N)/N = B(G/N).If The application of Corollary 2.8 for H = G gives the following result, that was also proved by Shchukin (see [19], Theorem 2).

Corollary 2.9 If G is any group, then B(G) is the largest normal subgroup of G admitting an ascending G-invariant series with abelian factors.
It follows in particular from the above corollary that B(B(G)) = B(G) for every group G. Let H and S be the full replete subcategories of the semi-abelian category Grp whose objects are all hyperabelian groups and all groups with no nontrivial abelian normal subgroups, respectively.The pair (H, S) is not a torsion theory (in the sense of [9]), because every group can be embedded in a simple nonabelian group and so there are plenty of nonzero homomorphisms from groups in H to groups in S.
Combining Theorem 2.6 and Corollary 2.8, we obtain the following characterization of hyperabelian groups in terms of spectrum.

Corollary 2.10 A normal subgroup H of a group G admits an ascending G-invariant series with abelian factors if and only if Spec
Let H be a normal subgroup of a group G.It clearly follows from the above corollary that if Spec G (H ) = ∅, then Spec(H ) = ∅.On the other hand, the converse of this fact does not hold in general.In fact, Philip Hall constructed a group G containing a nonabelian minimal normal subgroup M such that G/M is metabelian and M is hyperabelian (see [17], Part 2, p.81), so G is not hyperabelian, and Spec(M) = ∅ while {1} is the unique G-prime subgroup of M.
Let H be a normal subgroup of a group Proof Suppose first that all proper normal subgroups of G are prime and let H, K be arbi- It is easy to prove that a proper normal subgroup P of a finite group G is prime if and only if G/P contains a unique minimal normal subgroup and this is nonabelian.This is far from being true in the infinite case, as shown by the following result.Theorem 2. 16 There exists an infinite group G all of whose proper normal subgroups are prime and form a descending chain of type ω + 1.In particular, G has no minimal normal subgroups.
Proof Define a sequence of groups {G n } n∈N , each contained in the following one, as follows.Set G 0 = {1} and suppose that a group G n has been constructed for some nonnegative integer n.Consider the standard wreath product G n+1 = Alt(n + 4) G n with the natural embedding of G n into G n+1 , and denote by G the direct limit of the sequence of groups G n (for the definition and the main properties of standard wreath products see [17], Section 6.2).For each positive integer n, let B n be the base group of G n and put In this way, we have defined a descending chain of normal subgroups But G/K m−1 G m−1 and so it follows from the minimal choice of m that N = K m−1 .Thus G has no minimal normal subgroups, while an application of Theorem 2.15 yields that all proper normal subgroups of G are prime.
Notice that McLain constructed in [16] a periodic locally soluble group M whose lattice of normal subgroups is isomorphic to that of the group constructed above.Of course, all chief factors of M are abelian, but M has no non-trivial abelian normal subgroups, so that the identity subgroup is the only prime subgroup of M.
In this context, we also mention that Leinen [13] proved that, for every prime p, the lattice of normal subgroups of any existentially closed locally finite p-group G is a chain; moreover, the set of all chief factors of G (ordered in the obvious way) is dense and without endpoints.In particular, if G is countable, N (G) is order-isomorphic to R ∪ {±∞}, and in this isomorphism Spec(G) corresponds to R \ Q ∪ {−∞}.
We conclude this section by observing that a theory characterizing hypercentrality can be developed in a similar way to what we have done above.In what follows, we only give an idea on how to do this.
We call a normal subgroup N of a group G weakly prime if X ≤ N whenever X is normal in G and [X, G] ≤ N .Obviously, any prime subgroup is weakly prime.It is also clear that a normal subgroup N is weakly prime if and only if ζ(G/N) = {1} and hence all normal subgroups of a group G are weakly prime if and only if every homomorphic image of G has a trivial centre.
Let G be a group.We say that an element x of G is a Černikov element if for all sequences x = x 0 , x 1 , . . ., x n , . . .and y 0 , y 1 , . . ., y n , . . . of elements of G such that x n+1 = [x n , y n ] there is a nonnegative integer m such that x n = 1 for each n ≥ m.S.N.Černikov proved that a group is hypercentral if and only if all its elements are Černikov (see for instance [17] Part 1, Theorem 2. 19).If we denote by C(G) the set of all Černikov elements of G, then the previous result can be stated as follows: G is hypercentral if and only if C(G) = G.Lemma 2.17 Let G be a group and let x be a nontrivial Černikov element of G. Then

Theorem 2.18 If G is any group, then C(G) is the intersection of all weakly prime subgroups of G.
The above result shows in particular that C(G) is always a subgroup, but one can actually say more.

The Spectrum of a Group and its Topology
Let H be a normal subgroup of a group G.The Zariski topology on the spectrum Spec G (H ) is the topology whose closed subsets are the sets The topological space Spec G (H ) turns out to be a T 0 sober topological space (see [6], Lemma 2.6).Moreover, it follows from results in [6], that the partially ordered set of all G-semiprime subgroups of H is a complete distributive lattice, because it is isomorphic to the lattice of all open subsets of the topological space Spec G (H ).In this section we study Spec G (H ) from the topological point of view, for instance determining whether it is compact or not, studying its bases of open subsets, checking whether the intersection of two compact open subsets is compact, and so on.
Our first result is necessary in order to understand the relation between Spec(G) and Spec G (H ).

Lemma 3.1 Let H be a normal subgroup of a group G and let N be a non-triv
Let P be any G-prime subgroup of H such that P ∩ N = {1}.Then Let G be a group and let N ≤ H be normal subgroups of G. Using the above lemma, we can associate to each X ∈ Spec G (N ) a G-prime subgroup ϕ(X) of H by putting ϕ(X)/X = C H/X (N/X).The map ϕ is actually a bijection between Spec G (N ) and Spec G (H )\V G H (N ), since it is easy to check that the restriction map is the inverse of ϕ.In order to prove that ϕ is a homeomorphism, take a G-invariant subgroup L of H and take P ∈ V G H (L). If P does not contain N , then ψ(P ) Thus closed sets correspond in the bijection ϕ and ϕ is a homeomorphism.
Clearly, the canonical projection H → H/N induces a homeomorphism between the closed subset V G H (N ) of Spec G (H ) and Spec G/N (H /N ).We have thus proved the following theorem.
Theorem 3.2 (see also Theorems 2.1 and 2.2 of [1], and Proposition 3.1 of [8]) Let N ≤ H be normal subgroups of a group G. Then: Corollary 2.7 shows that B(G) is the intersection of all prime subgroups of G.We introduce now the dual concept, denoting by coB(G) the subgroup generated by all primes of G, with the convention that coB(G) = {1} whenever Spec(G) = ∅.Of course, if G has prime subgroups, then The following proposition is now easy to prove.

Proposition 3.3
Let G be a group.
(1) If G has a greatest prime subgroup P , then coB(G) = P , B G/P is trivial and

and G does not have a greatest prime subgroup, then
Spec G/coB(G) is empty and Spec G (coB(G)/B(G)) is homeomorphic to Spec(G).
For a commutative ring R there is a correspondence between ring direct product decompositions R = R 1 × R 2 , clopen subsets of Spec(R), and idempotents of the ring R (see [2], Exercise 22, pp.[13][14].For instance, Spec(R 1 × R 2 ) is homeomorphic to the disjoint union Spec(R 1 ) ∪ Spec(R 2 ) of the two topological spaces Spec(R 1 ) and Spec(R 2 ).We now see a corresponding result for spectra of groups.
follows that either G 1 ≤ P or G 2 ≤ P , and hence either P = G 1 × P 2 for some prime subgroup P 2 of G 2 or P = P 1 × G 2 for some prime subgroup P 1 of G 1 .This proves that there is a canonical bijection defined by ϕ(P 1 × G 2 ) = P 1 for every prime subgroup P 1 of G 1 and ϕ(G 1 × P 2 ) = P 2 for every prime subgroup P 2 of G 2 .In order to prove that ϕ is a homeomorphism, notice that a basis of closed subsets for Spec(G) is given by the closed sets where N 1 and N 2 are the projections of N onto G 1 and G 2 , respectively.Notice also that the spectrum of a direct product of two groups may be decomposed in distinct ways as the disjoint union of two clopen subsets.To see this, it is enough to consider the direct product G = A × B × C × D, where A and B are simple non-abelian groups, and C and D are non-trivial abelian groups; in this case, we have We now show that the above theorem can be useful in the study of a group G whose spectrum is the disjoint union of two clopen subsets, since, in this case, direct products are naturally involved in the structure of G, up to hyperabelian sections.Lemma 3.5 Let G be a group and N 1 , N 2 be normal subgroups of G. Then: and only if there is no prime subgroup P of G such that P ≥ N 1 and P ≥ N 2 , or, equivalently, N 1 N 2 is not contained in P .Thus (b) Obviously, V (N 1 ) ∪ V (N 2 ) = Spec(G) implies that for every prime subgroup P of G, either P ≥ N 1 or P ≥ N 2 .In both cases, P ≥ N 1 ∩ N 2 and hence B(G) ≥ N 1 ∩ N 2 by Corollary 2.7.Conversely, suppose N 1 ∩ N 2 ≤ B(G) and let P be any prime subgroup of G. Then Notice that, in the above statement, the conditions in (a) are also equivalent to Spec(G/N 1 N 2 ) = ∅, while the conditions in (b) are equivalent to Spec G (N 1 ∩ N 2 ) = ∅.

Corollary 3.6
Let G be a group and N 1 , N 2 be normal subgroups of G. Then Spec(G) = V (N 1 ) ∪ V (N 2 ) if and only if the factor group G/N 1 N 2 is hyperabelian and For any ring R, set U(r) = {P ∈ Spec(R) | r / ∈ P } for all r ∈ R. The fundamental formula U(r) ∩ U(s) = U(rs) for all r, s in a commutative ring R is the reason why spectra of commutative rings are easy to handle with.In the case of a noncommutative ring R with identity, the above formula becomes for all r, s ∈ R.This fact has several implications.
(1) The sets {U(r) | r ∈ R} is a basis for the open sets (not only a subbasis, which is a considerable simplification).
Proposition 3.9 Let G be a group with Spec(G) = N (G) \ {G}.
(a) The topological space Spec(G) is sober, the intersection of two compact open sets is compact and the compact opens form a basis for the topology.(b) The topological space Spec(G) is spectral if and only if it is compact, if and only if G has a maximal normal subgroup.
It follows in particular from Proposition 3.9 (b) that if G is the example constructed after Theorem 2.15, then Spec(G) is a spectral space.The following statement (for a limit ordinal α) provides examples of groups G in which all proper normal subgroups are prime, although the topological space Spec(G) is not compact.Theorem 3.10 For each ordinal number α there exists a group G such that N (G) is similar to α + 1 and all proper normal subgroups of G are prime.
Proof Let X be a set of cardinality ℵ α and consider the symmetric group Sym(X).For each ordinal β ≤ α + 1 the set H β of all permutations on X whose support has cardinality < ℵ β is a normal subgroup of Sym(X).Clearly, H α+1 = Sym(X) and it is well known that where FAlt(X) denotes the finitary alternating group on X, and that the factor group H β+1 /H β is simple non-abelian for each β < α + 1.In particular, N (H α+1 /H 0 ) is well ordered of order type at least α + 1, and it follows from Theorem 2.15 that, for each β ≤ α + 1, all proper normal subgroups of H β /H 0 are prime.In order to complete the proof, it is now enough to choose β such that N (H β /H 0 ) is similar to α + 1.
Finally, we remark that the construction of the spectrum Spec(G) of a group G is not functorial at all.To see this, consider the group Alt(8), its Klein four-group V = (1 2)(3 4), (5 6) (7 8) and the inclusion V → Alt(8); since Spec(V ) = ∅ while | Spec(Alt(8))| = 1, we cannot find any contravariant functor Grp → Top mapping groups to their spectra.A similar consideration, replacing Alt(8) by Alt(8) × V , shows that there is no covariant functor Grp → Top mapping groups to their spectra.

m-Systems
Let R be a ring.Recall that a nonempty subset S of R is an m-system if, for all x, y ∈ S, there is an element r ∈ R such that xry ∈ S (see [15], or [12], pp.166-167).Of course, m-systems generalize multiplicatively closed subsets of a ring.It is well known that msystems and prime ideals of a ring are strictly connected.In fact, for a ring R, the following statements hold.
(1) An ideal P of R is prime if and only if R \ P is an m-system.
(2) If S is an m-system and P is an ideal of R which is maximal with respect to the condition P ∩ S = ∅, then P is prime.
By a standard application of Zorn's lemma, it follows from (2) that any m-system S with 0 / ∈ S determines prime ideals.
The aim of this section is to introduce and study the basic properties of a corresponding concept for groups, which can be useful in order to find prime subgroups.We say that a nonempty subset S of a group G is an m-system if, for any a, b ∈ S, there exists g ∈ G such that [a g , b] ∈ S.
Our next two results prove that m-systems in groups work similarly to m-systems in rings.The first of them is an immediate consequence of Lemma 2.3(1).
Let G be a group.A subset S of G is called n-system if, for every a ∈ S, there is g ∈ G such that [a g , a] ∈ S (see [12, p. 167]).Clearly, every m-system is an n-system, and it follows from the definitions that an element x of G is a Baer element if and only if the identity 1 belongs to any n-system of G containing x. Thus G is hyperabelian if and only if all its nonempty n-systems contain 1. Conversely, let S be an m-system of G containing x such that S ∩ N = ∅.Then there exists a normal subgroup P ≥ N of G which is maximal with respect to the property P ∩ S = ∅.Then P is prime by Proposition 4.2 and x ∈ / P , so that x ∈ / B(G : N).

Corollary 4.4
Let G be a group.Then B(G) is the set of all elements x ∈ G such that 1 belongs to every m-system of G containing x.In particular, G is hyperabelian if and only if every m-system of G contains 1.
Arguments similar to those employed for prime subgroups (see Lemma 4.1 and Proposition 4.2) show that: (1) a normal subgroup N of G is semiprime if and only if its complement G\N is an n-system; (2) if S is an n-system of G and N is any normal subgroup of G which is maximal with respect to the condition N ∩ S = ∅, then N is semiprime.
In the case of rings, every n-system is a union of m-systems.This is not the case for groups, as the following result shows.Theorem 4.5 There exists a group G containing a subset S which is an n-system of G but cannot be obtained as union of m-systems of G.
Proof Consider the free group F on the alphabet . . ., a −2 , a −1 , a 0 , a 1 , a 2 , . . .and let g be the automorphism of F mapping a i to a i+1 for each integer i.Put G = g F .Then S = a 0 , [a g 0 , a 0 ] = [a 1 , a 0 ], [a 1 , a 0 ] g 2 , [a 1 , a 0 ] = [a 3 , a 2 ], [a 1 , a 0 ] , . . . is an n-system of G. Let us prove that S cannot be decomposed into a union of m-systems.
To see this, assume for a contradiction that a 0 belongs to an m-system S 0 ⊆ S. Of course, there is x ∈ G such that [a x 0 , a 0 ] ∈ S 0 , and hence {a 0 } ⊂ S 0 .Let us order the set S accordingly to the way we have written the elements, that is, Declarations The authors declare that data supporting the findings of this study are available within the article.
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Corollary 2 . 8
If H is any normal subgroup of a group G, then B H (G) is the largest Ginvariant subgroup of H admitting an ascending G-invariant series with abelian factors.Proof Since B H (G) = H ∩ B(G), it is enough to prove the statement when H = G.Let N be any proper G-invariant subgroup of B(G) and let x be a Baer element of G which is not in N .Then xN is a Baer element of G/N and so x G N/N contains an abelian nontrivial Ginvariant subgroup by Lemma 2.5.Therefore B(G) admits an ascending G-invariant series with abelian factors.
has no abelian nontrivial G-invariant subgroups and hence B(G) is the largest normal subgroup of G with an ascending G-invariant series with abelian factors.
to the base group of G n , and n K n = {1}.Let N be any proper nontrivial normal subgroup of G and consider the smallest positive integer m such that N ∩ G m = {1}.Then N contains the base group B m of G m , because B m is the unique minimal normal subgroup of G m .By the same reason, B n ≤ N for each n ≥ m and hence

Lemma 4 . 1 AProposition 4 . 2
normal subgroup N of a group G is prime if and only if its complement G \ N is an m-system.Let S be an m-system of a group G and let N be any normal subgroup of G which is maximal with respect to the property N ∩ S = ∅.Then N is prime.Proof Assume N is not prime, so by Lemma 2.3(1) there are a, b ∈ G \ N such that [ a G , b G ] ≤ N .Clearly, both a G N and b G N properly contain N and hence, by the maximality of N , there exist s, t ∈ S with s ∈ a G N and t ∈ b

Lemma 4 . 3
Let N be a normal subgroup of a group G. Then B(G : N) is the set of all elements x of G such that S ∩ N = ∅ for every m-system S containing x. Proof Let x be an element of G.If x / ∈ B(G : N), then the coset xN does not belong to B(G : N)/N = B(G/N), and so by Corollary 2.7 there exists a prime subgroup P of G containing N such that x / ∈ P .Then G \ P is an m-system of G by Lemma 4.1, contains x and is disjoint from N .
a 2 ], [a 1 , a 0 ] < . . .Now, let u be the least element of S \ {a 0 } that belongs to S 0 .Then [a y i , u] ∈ S 0 for two suitable elements i ∈ Z and y ∈ F .Clearly, [a y i , u] = u and so u < [a y i , u].If A is the set of letters involved in u, we may assume a j = 1 for every a j / ∈ A ∪ {a i }.It follows that v = 1, whenever v ∈ S and v > u; in particular, [a y i , u] = 1, which is obviously impossible.Funding Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.

Theorem 2.15 Let
H , we speak of G-semiprime subgroups instead of G/K-semiprime subgroups.Corollary 2.11 Let H be a normal subgroup of a group G. Then B H (G) is the intersection of all G-semiprime subgroups of H . Proof Of course, we may assume that the set S of all G-semiprime subgroups of H is nonempty, so that the intersection S of all members of S is a G-semiprime subgroup, which is contained in B H (G) by Theorem 2.6.Then H/S has no non-trivial abelian G-invariant subgroups and hence it follows from Corollary 2.8 that B H (G) = S.As a consequence of Corollary 2.11 and Theorem 2.6 we have the following results.Corollary 2.12 Let G be a group.Then the intersection of all prime subgroups of G coincides with the intersection of all semiprime subgroups.Corollary 2.13 Let G be a group.A normal subgroup N of G is semiprime if and only if it is the intersection of all prime subgroups containing N .Let H be a normal subgroup of a group G and let N be a proper G-semiprime subgroup of H . Then there exists a G-prime subgroup of H containing N .Proof As the trivial subgroup of H/N is G-semiprime, it follows from Corollary 2.11 that the intersection of all G-prime subgroups of H/N is trivial as well, and hence N lies in some G-prime subgroup of H . Corollary 2.10 characterizes groups with an empty spectrum.On the opposite, we study now the other extreme case in which Spec(G) = N (G) \ {G}.Part of the next statement already appears in [20, Behauptung 2.1].G be a group.Then all proper normal subgroups of G are prime if and only if the lattice N (G) is a chain and H = H for every normal subgroup H of G.