## Structure and Representations of Q-groups by Dennis Kletzing By Dennis Kletzing

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Let T be an involution of G. Then T splits the set I into the following two subsets: i+ = { T' c I I TT' ~ I } I -T = {,~'~I I TT'~I}, T 29 IT I+=m TI: TI T Y TI Proposition 29. Proof. The number of involutions in G is odd. Let X stand for the character of the permutation representation of G acting on I by conjugation. Then, for any involution T, it follows that X(T) : # involutions in C(T). Now, m' s I+T if and only if T' conmutes with T and T' ~ T (T £ I+T since T 2 = i £ I). Hence + z. Since TI: = I+T' it follows that T defines a fixed-point free bijection of order two on the set I+T and hence II:I is even.

Let S ~ T n and T ~ U m. Then S ~ unmwhence K(S) < K(U). It now follows that 2(G) is partially ordered by <. // To show howthis partial order relates to the standard partial order on the classes of subgroups, let S(G) stand for the set of conjugacy classes of subgroups of G. S(G) is partially ordered by setting K(H) <_* K(L) whenever H is conjugate to 49 a subgroup of L. Let S(G)cyclie and define a mapping f:~(G) = { K(H) s S(G) I H is cyclic } ) S(G) cyclic by setting f(K(S)) = K((S)). If G is a Q-group, the map f is an order-preserving isomorphism.

The p-class decompositions corresponding to these components are as follows: K2(1) = K(sr) + K(r 3) \$ K(s) \$ K(1) K2(r 2) = K(r 2) + K(r) K3(sr) = K(sr) K3(1) = K(1) + K(r 2) K3(r3) = K(r 3) \$ K(m) K3(s) = K(s). This conpletes the discussion of the partially ordered set (~(G), <) and the graph ~*(G). ChapteF 2. Constructions of q-Groups In the previous chapter it was shown that the property of being a Q-group is preserved by forming direct products and quotients. Thus, these elementary grouptheoretic constructions produce Q-groups from Q-g~ups.