0
votes
0answers
7 views

maximal abelian subgroup

let M(G) denote the set of orders of maximal abelian subgroups of G. If M(G) = M(H), for some group H then what can we say about the prime numbers that divide the order of each group G and H?
2
votes
1answer
186 views

The special subgroups of a finite abelian group of rank two

Let $G=\langle a_{1}\rangle\times\langle a_{2}\rangle$ such that $|a_{i}|=2^{k_{i}}$ and $k_{1}>k_{2}$ and $H$ be a subgroup of $G$ that there exists an automorphism of $G$ such that fix only ...
6
votes
1answer
469 views

Automorphism group of a finite group

I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = ...
1
vote
0answers
153 views

How many subgroups of order $\prod_{1}^{n} p_{i}^{n_{i}}$ are there in the finite product of cyclic groups?

All of the following ${p_{i},q_{i}}$are prime numbers, ${n,m,k}$ are pre-assigned integers. Consider the product of cyclic groups $\prod_{1}^{n} \mathbb Z_{p_{i}^{n_{i}}}$ then we asked the question: ...
2
votes
3answers
330 views

Finite / uniquely divisible abelian groups

Is there any counter example for the following statement? STATEMENT: Let $0 \to F \to A \to Q \to 0$ be a short exact sequence of abelian groups. Assume that $F$ is a finite group, and $Q$ is a ...
2
votes
1answer
495 views

Linear algebra of finite abelian groups

If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...
3
votes
1answer
295 views

On the existence of a direct summand containing a fixed subgroup

Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups that they generate are in direct sum $\langle g_1 \rangle \oplus \ldots \oplus \langle g_n \rangle$. Is it ...
0
votes
1answer
475 views

Maximal subgroups of a finite p-group

I want to prove the following: Let $G$ be a finite abelian $p$-group that is not cyclic. Let $L \ne {1}$ be a subgroup of $G$ and $U$ be a maximal subgroup of L then there exists a maximal subgroup ...
3
votes
1answer
515 views

Cyclic subgroups of finite abelian groups

I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one. Are there simple formulas if one ...
2
votes
1answer
538 views

Minimal generation for finite abelian groups

Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups: 1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$, 2) With orders that are ...