1
vote
2answers
30 views

Example of a subgroup of index two which contains a non square element

If a finite group contains a subgroup H of index two, then every element of the group which is a square belongs to H. Is there a (simple) counterexample showing that not all the elements of H are ...
4
votes
1answer
58 views

Groups of order $p^aq^br^c$ containing elements of order $pq$, $qr$, and $pr$, but not $pqr$

Let $G$ be a solvable group of order $p^aq^br^c$ (for distinct prime $p,q,r$) containing elements of orders $pq$, $qr$, and $pr$, but no element of order $pqr$. Furthermore, assume that $G$ is ...
3
votes
1answer
83 views

If all the centralizers are finite, then does it follow that the group is finite?

As far as I have heard, centralizers play an important role in the theory of groups. My question arises from curiosity and the desire to understand how much control centralizers have over the group. ...
10
votes
0answers
149 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
3
votes
1answer
306 views

Counter examples in group theory

Let $G$ be a finite group with normal subgroups $N_{1}$ and $N_{2}$. Find counter examples to the following statements 1) If $N_{1}\cong N_{2}$ then $G/N_{1}\cong G/N_{2}$ 2) If $G/N_{1}\cong ...
1
vote
3answers
141 views

Formula for Product of Subgroups of $\mathbb Z$, Problem

What is the product of $\mathbb{Z}_2$ and $\mathbb{Z}_5$ as subgroups of $\mathbb{Z}_6$? Since $\mathbb{Z}_n$ is abelian, any subgroup should be normal. From my understanding of the subgroup product, ...
27
votes
9answers
5k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
1
vote
1answer
146 views

On finite groups whose center is elementary abelian group

Let $G$ be a finite 2-group such that $Z(G)$ is elementary abelian 2-group ($\mid Z(G)\mid\geq 4$) and $Inn(G)$ is of order 4. Then prove that there exists an $\alpha\in Aut(G)$ such that ...
9
votes
1answer
350 views

Is there a counterexample to this weakened converse of Hall's theorem?

Suppose that a finite group $G$ contains a Hall $\{p,q\}$-subgroup for every pair of prime divisors $p,q$ of $|G|$. Does it follow that $G$ is solvable?
3
votes
1answer
138 views

On groups whose center has odd order

Let $G$ be a finite group such that $Z(G)$ is of odd order and $Inn(G)$ is of even order. Then prove $G\simeq Z(G)\times N$, such that $N$ is a subgroup of $G$ where $N\simeq Inn(G)$. Thank you!
1
vote
1answer
269 views

A question about intersection of center and commutator subgroup

Let $G$ be a finite group such that $G'\cap Z(G)\neq 1$. Suppose also that $G'$ is an elementary abelian $p$-group; $G'\nleq Z(G) $; $(G/Z(G))'$ is a minimal normal subgroup of $G/Z(G)$. Can we ...
2
votes
2answers
261 views

Let $G$ be a group of order $56$. Then which of the following are true

Let $G$ be a group of order $56$. Then which of the following are true All $7$-sylow subgroups of $G$ are normal All $2$-Sylow Subgroups of $G$ are normal Either a $7$-Sylow subgroup or a ...
22
votes
1answer
2k views

Isomorphic quotients by isomorphic normal subgroups

In this recent question, Iota asked if, given a finite group $G$ and two isomorphic normal subgroups $H$ and $K$, it would follow that $G/H$ and $G/K$ are isomorphic. This is not true (a simple ...
14
votes
4answers
2k views

Derived subgroup where not every element is a commutator

Let $G$ be a group and let $G'$ be the derived subgroup, defined as the subgroup generated by the commutators of $G$. Is there an example of a finite group $G$ where not every element of $G'$ is a ...