A lattice-theoretic characterization of pure subgroups of Abelian groups

Let G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice L(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}(H)$$\end{document} embeds in L(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}(G)$$\end{document}. It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.


Introduction
The aim of this short note is to define a sublattice embedding property which, into the universe of subgroup lattices of abelian groups, is satisfied precisely by all the elements corresponding to pure subgroups.Recall that a subgroup H of an abelian group G is said pure in G if G n ∩ H = H n for all positive integers n, or, equivalently, if every h ∈ H having a nth root in G admits a nth root in H (for the main properties of these subgroups see for instance [3]).This work is motived by that of Černikov in [1] (see also the recent survey [2]), where he describes the structure of abelian groups whose pure subgroups admit a complement.An immediate consequence of our main result is that any projectivity between abelian groups preserves the property considered by Černikov.
Notice that all results are carried out in a local nilpotent context for a general definition of pure subgroup.In particular, the final result of the paper (Corollary 2.6) shows that pure subgroups are characterized by a lattice-theoretic property even in the universe of groups whose subgroup lattice is isomorphic to that of a nilpotent group.
Most of our notation is standard and can be found in [3] and in [4].

The main result
In order to make things clear we split the main definition into smaller pieces.Let L be a lattice with maximum 1 and minimum 0.
] is a finite chain we say that c is primary cyclic.
A well known consequence of a theorem of Ore is that a subgroup H of a group G is cyclic if and only if it is a cyclic element of the subgroup lattice L(G) of G (see for instance [4,Corollary 1.2.4]).It also follows that H is a primary cyclic element of L(G) if and only if H is a cyclic p-group for some prime p.It is clear that x properly splits in x ∨ ϕ(x) if and only if ϕ(x) does so.This remark can be easily employed to show that the relation defined by "having the same cyclic structure" is an equivalence relation.In terms of subgroup lattices of locally nilpotent groups, this relation "identifies" those cyclic subgroups which are isomorphic.

Lemma 2.1 Let C and D be finite cyclic subgroups of a locally nilpotent group G. Then C and D have the same cyclic structure as elements of L(G) if and only if they are isomorphic.
Proof Suppose first C is isomorphic to D and let ϕ be an isomorphism.Then ϕ induces a lattice isomorphism between L(C) and L(D).If X is a subgroup of C with totally ordered subgroup lattice, then it is a p-subgroup for some prime p.Thus X , ϕ(X ) is a p-group and so X does not properly split in it (see [4,Theorem 1.6.5]).
Conversely, assume C and D have the same cyclic structure and let be a lattice isomorphism with the property prescribed by the definition (c).Let p ∈ π(C) and let C p be the Sylow p-subgroup of C. Now, Corollary 1.2.8 of [4] shows that ϕ(C p ) is a Sylow q-subgroup of D for some prime q.If p = q, then C p , ϕ(C p ) = C p ×ϕ(C p ), against the property of ϕ.Thus p = q and π(C) ⊆ π(D) by the arbitrariness of p ∈ π(C).By symmetry, π(C) = π(D).Finally, Corollary 1.2.8 of [4]  The proofs of the above lemma and its corollary make it clear that in a locally nilpotent group any projectivity between two its cyclic subgroups with the property prescribed by definition (c) comes from an isomorphism.
We are now in a position to give our main definition.
(d) An element x ∈ L is said to be pure in L if for every infinite or primary cyclic element c of L such that x ∧ c = 0, there is a cyclic element y of L with x ∧ c ≤ y ≤ x and such that [y/x ∧ c] has the same cyclic structure of Before proving our main result we generalize the definition of pure subgroups.A subgroup H of a group G is pure in G if, for all positive integers n, every h ∈ H having a nth root in G admits a nth root in H .

Lemma 2.3 Let H be a pure subgroup of a group G. Then H is a pure element of L(G).
Proof Let C = c be a cyclic subgroup of G which is either infinite cyclic or of prime power order, and such that c n = H ∩ C = 1 for some smallest positive integer n.The purity of H in G yields the existence of h ∈ H with h n = c.The hypothesis on C shows that h / c n and c / c n are cyclic groups of order n, so they are isomorphic and hence they share the same cyclic structure by (argue similarly to the first part of the proof of Lemma 2.1).

Theorem 2.4 Let H be a subgroup of a locally nilpotent group G. Then H is a pure element of L(G) if and only if H is a pure subgroup of G.
Proof By Lemma 2.3 we may assume that H is a pure element of L(G).Suppose by contradiction that H is not pure in G, so there is a smallest positive integer n such that there are g ∈ G and h ∈ H with g n = h, but there is no h 1 ∈ H with h n 1 = h.Since G is locally nilpotent, it is possible to assume that g is either infinite cyclic or of order a power of a prime p; in the latter case, the minimality of n shows that n itself is a power of p.Moreover, where m can be chosen dividing n, so again the minimality of n shows that m = n.Now, the definition of pure element yields the existence of k ∈ H such that h ≤ k , while k / h and g / h have the same cyclic structure.It follows from Lemma 2.1 that the latter two finite cyclic groups are actually isomorphic.This easily yields that h has a nth root in H , a contradiction proving the result.

Corollary 2.5
Let G 1 , G 2 be locally nilpotent groups and let ϕ be a projectivity from Corollary 2.6 Let ϕ be a projectivity between the locally nilpotent group G 1 and the group G 2 .Then H is a pure subgroup of G 1 if and only if H ϕ is a pure subgroup of G 2 .
Proof If H ϕ is a pure subgroup of G 2 , then it is also a pure element of L(G 2 ) by Lemma 2.3.Thus H is a pure element of L(G 1 ) and so a pure subgroup of G 1 by Theorem 2.4.
Suppose H is a pure subgroup of G 1 and, by contradiction, that H ϕ is not pure in G 2 .Then there is a smallest positive integer n such that there are g 2 ∈ G 2 and h 2 ∈ H ϕ with g n 2 = h 2 , but there is no x ∈ H ϕ with x n = h 2 .Put H 2 = h 2 and suppose first that either g 2 is infinite cyclic or g 2 has prime power order p m .Then, by minimality of n, we have that . Moreover, by hypothesis, there is a cyclic subgroup Assume now that g 2 is cyclic of order a power of p; in particular, g 2 ϕ −1 is cyclic of order a power of a prime q.Since g have the same cyclic structure, then the order of C 1 /H ϕ −1 2 is a power of q (see Lemma 2.1) and so C 1 is cyclic of order a power of q.It follows that c 2 has order a power of the prime p and so we have that c 2 /H 2 and g 2 /H 2 are isomorphic, a clear contradiction.
Let's now deal with the case in which g 2 is infinite.Since is nilpotent and finite, it follows that c 2 , g 2 /H 2 is a direct product Finally, suppose H 2 has finite order and notice that the minimality of n shows that π(n) ⊆ π(H 2 ).Let p be a prime, denote by n p the maximum power of p dividing n, by h 2 ( p) the Sylow p-subgroup of H 2 and by g 2 ( p) an element of g 2 such that g 2 ( p) n p = h 2 ( p).What we have already proved shows that there is a p-element c 2 ( p) of H 2 such that c 2 ( p) n p = h 2 ( p).The subgroup generated by all these c 2 ( p) has a subgroup lattice which is lattice-isomorphic to that of a finite nilpotent group, so it is a direct product of primary groups and P-groups with relatively prime orders (see for instance [4, Exercise 2.2.7]).Since the socles of the subgroups c 2 ( p) 's commute with each other, it easily follows that the c 2 ( p)'s belong to distinct direct factors and so they also commute with each other.It is now clear that h 2 admits an nth root in H , this is our final contradiction.
(b) Let c ≤ d be elements of L. Then c is said to split in d if [d/0] L × [c/0] for some lattice L .If [c/0] [d/0] or c = 0, we say that c improperly splits in d, otherwise we say that it does so properly.(c) Let c, d ∈ L. We say that c and d have the same cyclic structure if there exists a lattice isomorphism ϕ : [c/0] −→ [d/0] such that x does not properly split in x ∨ ϕ(x) whenever [x/0] is a chain.
yields that C and D are isomorphic.Proof By Lemma 2.1, it only remains to prove that if C and D are infinite cyclic subgroups of G, then they have the same cyclic structure as elements of L(G).However, it is clear that L(Z) has no subgroup X such that L(X ) is a chain and so the condition in definition (c) is trivially satisfied.
H 2 of primary groups and P-groups with relatively prime orders (see for instance [4, Exercise 2.2.7]).By definition (c), the Sylow (primary) subgroups of c 2 /H 2 and g 2 /H 2 must correspond each other and must lie in the same direct factor K i /H 2 .If K i /H 2 is a primary group, then these Sylow subgroups are trivially isomorphic; if K i /H 2 is a P-group, then they cannot have distinct orders since otherwise one would act non-trivially on the other contradicting the fact that H 2 ≤ Z ( c 2 , g 2 ).Therefore, in any case corresponding Sylow subgroups are isomorphic and hence such are c 2 /H 2 and g 2 /H 2 , again a contradiction.