Minimal generation for finite abelian groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:46:42Z http://mathoverflow.net/feeds/question/82603 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82603/minimal-generation-for-finite-abelian-groups Minimal generation for finite abelian groups Calc 2011-12-04T08:35:39Z 2011-12-07T11:28:53Z <p>Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups:</p> <p>1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$,</p> <p>2) With orders that are powers of not necessarily distinct primes $p_1^{\alpha_1}, \ldots, p_n^{\alpha_n}$.</p> <p>Is it true, and how can one prove that the cardinality $c$ of any minimal generating set for $G$ satisfies $k \leq c \leq n$ (I am most concerned about the second inequality)? Here minimal means irredundant.</p> http://mathoverflow.net/questions/82603/minimal-generation-for-finite-abelian-groups/82618#82618 Answer by Geoff Robinson for Minimal generation for finite abelian groups Geoff Robinson 2011-12-04T13:08:28Z 2011-12-07T11:28:53Z <p>Note that $n$ is the sum over prime divisors $p$ of $|G|$ of the minimal number of generators of the distinct Sylow $p$-subgroups of $G.$ The sizes of all minimal generating sets of a finite $p$-group are the same by properties of the Frattini subgroup. Use of the Frattini subgroup helps to prove the leftmost inequality: take a prime $p$ which divides $d_1 .$ Then a Sylow $p$-subgroup of $G$ can't be generated by fewer than $k$ elements, so $G$ itself certainly can't be generated by fewer than $k$ elements, as each Sylow $p$-subgroup of $G$ is a homomorphic image of $G.$ On the other hand, take a minimal generating set $S$ for $G$ of maximal cardinality, and minimize the sum of the orders of elements of $S$ subject to that. Then each element of $S$ must have prime power order, for if $s \in S$ has order divisible by more than one prime, then we may write $s = t + u$ where $t$ and $u$ have coprime orders (each greater than one) whose product is the order of $s$. Then <code>$(S \backslash \{ s \}) \cup \{t,u\}$</code> is still a minimal generating set for $G,$ contradicting the maximality of the cardinality of $S.$ The fact that $S$ is a minimal generating set means that if we now collect the elements of $S$ whose orders are powers of a fixed prime $p$, we must obtain a generating set for a Sylow $p$-subgroup of $G,$ and this must be minimal by the choice of $S$. Hence the cardinality of $S$ is at most $n,$ as defined above.</p>