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By Iyanaga S. (ed.)

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1 generalizes immediately. 2 Proposition Let H be a nontrivial subgroup ofafinite group G with e denoting the identity. L( GIH) is the least positive integer n such that, for some subgroups HI , ... , Hm of G, we have n core(Hi) m H n m = {e} :L and i=1 IG:Hil = n. L(GIH) is induced by right multiplication of the cosets of the subgroups HI , ... , Hm . Consider now a semilattice E and suppose that S is a semilattice of groups where each group G e has an identity which may be identified with e. The multiplication in S may be reconstructed from the group multiplications, the semilattice multiplication and connecting homomorphisms

Then (if necessary) the primes PI , ... ,Pn may be rearranged so that p,( GIN) = m L pfi . i=l The idea of the proof is to deform N into a direct summand of G without altering p,( GI N) so that p,( GI N) = p,(N) and then use the result quoted earlier about minimal degrees of abelian groups. From the method one may construct a representation which realizes p,( GIN) . Before proving the theorem we give some easy lemmas. 2 Lemma Suppose that A, Band C are subgroups of G which are such that AnB = ABnC = {I}.

Definition is simply a way of moving elements of ~ from the purely abstract representation as members of r to the concrete representation as members of S. The algorithm modifies ~ , which is initially identical to r until it becomes isomorphic to n. Of course it is vital in practise to consider how much of ~ will actually have to be represented concretely at any stage. 1 THE ALGORITHM Starting conditions We begin the calculation with just the n required elements in Sand ds(i) = false, Ts(i) = 1-, o:(s(i») = l(i) and Ts(i) x = 1-, Vx EX, 1 :::; i :::; n.

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