On the Lie Ring of a Group of Prime Exponent

Abstract

Certain properties of a p-group G are reflected in its Lie ring L(G). For example, if the identity

$${x^p} = 1$$

(1.1)

holds in G, then the identities

$$pu = 0,$$

(1.2)

$$\;\underbrace {\left[ {\left[ { \ldots \left[ {u,v} \right], \ldots } \right],v} \right]}_{p - 1\,{\text{terms}}} = 0$$

(1.3)

hold in L(G). Thus, we have an isomorphism of graded Lie rings

$$L\left( {B\left( n \right)} \right) \cong \wedge \left( n \right)/\sum \left( n \right),$$

(1.4)

where B(n) is the n-generator free group of the variety of groups defined by (1.1) and Λ(n) the n-generator free Lie ring of the variety of Lie rings defined by (1.2) and (1.3).