3 - Finite groups

Summary

In this chapter, we denote the cardinality of a set F by |F|. Thus the order of a group G is denoted |G|. We recall that R(G), or just R, denotes the set of reversible elements of G, and R_g(G), or just R_g, denotes the set of elements h that satisfy hgh⁻¹ = g⁻¹. Also I(G), or I, denotes the set of involutions in G, and Iⁿ = {τ₁ … τ_n : τ_i ∈ I}.

Reversers of finite order

Proposition 3.1Suppose that elements g and h of a group G satisfy hgh⁻¹ = g⁻¹. If h has finite order, then either g is an involution or else the order of h is even. In the latter case, g is also be reversed by an element whose order is a power of 2.

Proof To prove the first assertion, suppose that the order of h is odd. Since each odd power of h reverses g, it follows that the identity 1 reverses g. Therefore g = g−1, so g is an involution.

Suppose now that h has even order 2^kq for positive integers k and q, where q is odd. Then h^q also reverses g, and this element has order 2^k. This proves the second assertion.

Corollary 3.2A finite group G contains a nontrivial reversible element if and only if G has even order. In fact, if G contains a nontrivial reversible element, then it contains a nontrivial involution.

Proof Suppose first that G has a nontrivial element g that is reversed by another element h. By Proposition 3.1, one of g or h has even order, so G also has even order.

Suppose now that G has even order. By partitioning G into subsets {g, g⁻¹} we see that G contains a nontrivial involution. Since involutions are reversible, all parts of the corollary have been accounted for.