By Lucchhini A.

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L1 . . lu )g −1 = (g(i1) . . g(ir ))(g(j1 ) . . g(js )) . . g(lu)) (even if the cycles are not disjoint). In other words, to obtain gαg −1 , replace each element in a cycle of α be its image under g. We shall now determine the conjugacy classes in Sn . By a partition of n, we mean a sequence of integers n1 , . . , nk such that 1 ≤ ni ≤ ni+1 ≤ n (all i) and n1 + n2 + · · · + nk = n. Thus there are 1, 2, 3, 5, 7, 11, . . partitions of 1, 2, 3, 4, 5, 6, . . respectively (and 1, 121, 505 partitions of 61).

28. (a) A simple group is indecomposable, but an indecomposable group need not be simple: it may have a normal subgroup. For example, S3 is indecomposable but has C3 as a normal subgroup. (b) A ﬁnite abelian group is indecomposable if and only if it is cyclic of prime-power order. Of course, this is obvious from the classiﬁcation, but it is not diﬃcult to prove it directly. Let G be cyclic of order pn , and suppose that G ≈ H ×H . Then H and H must be p-groups, and they can’t both be killed by pm , m < n.

Let P ∈ O, and consider the action by conjugation of P on O. This single G-orbit may break up into several P -orbits, one of which will be {P }. 8 that then Q = P . Hence the number of elements in every P -orbit other than {P } is divisible by p, and we have that #O ≡ 1 mod p. Suppose there is a P ∈ / O. We again let P act on O, but this time the argument shows that there are no one-point orbits, and so the number of elements in every P -orbit is divisible by p. This implies that #O is divisible by p, which contradicts what we proved in the last paragraph.

### A 2-generated just-infinite profinite group which is not positively generated by Lucchhini A.

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