### A cute group theory problem

Jan. 17th, 2014 04:51 pm**sasha_br**

Here is an elementary (but not completely trivial) fact from group theory that I didn't know. Let A be a finite abelian group. Let us define the trace of A (tr(A))

to be the sum of all n_i if A is isomorphic to the product of cyclic groups of sizes n_1,...,n_k, where each n_i is a prime power. Then A is a subgroup of S_n if and only if tr(A) is less than of equal to n.

to be the sum of all n_i if A is isomorphic to the product of cyclic groups of sizes n_1,...,n_k, where each n_i is a prime power. Then A is a subgroup of S_n if and only if tr(A) is less than of equal to n.