Articles of group theory

Subgroups of Abelian Groups

In the following problems, let $G$ be an Abelian group. 1) Let $H = \{ x \in G: x=y^{2} \text{ for some } y \in G \}$; that is, let $H$ be the set of all the elements of $G$ which have a square root. Prove that $H$ is a subgroup of $G$. (i). Let […]

Simplifying Relations in a Group

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$. As I have asked here, I am trying to use GAP to do some calculation with $K$. Now one problem is that there are too many relations. For example, all following […]

The center of a group with order $p^2$ is not trivial

Let $p$ be a prime and $G$ be a group of order $p^2$. Show that $Z(G)\neq 1$. Is there a proof of this nice fact that doesn’t use the class equation?

How to find a smallest group generator

Find the smallest generator for a group $\mathbb{Z}^{*}_{31}.$ It is not said, but I assume that it is about multiplication, because addition would be a trivial one (1 would be a generator).

Classifying groups of order 12.

I was trying to classify groups of order 12 and I ended up with 5 different groups: $\bullet$ $\Bbb{Z}_{12}$ $\bullet$ $\Bbb{Z}_2 \times \Bbb{Z}_6$ $\bullet$ $(\Bbb{Z}_2 \times \Bbb{Z}_2) \rtimes_{\alpha} \Bbb{Z}_3$ where $\alpha: (1,1) \rightarrow \bar{-1}$ $\bullet$ $(\Bbb{Z}_2 \times \Bbb{Z}_2) \times \Bbb{Z}_3$ $\bullet$ $\Bbb{Z}_3 \rtimes \Bbb{Z}_4$ where $\alpha$ sends the generator to $\bar{-1}$ I want to show […]

Classifying groups of order 21.

We have two sylow subgroups of orders 7 and 3. Let $n_3$ and $n_7$ denote the number of sylow subgroups for 3 and 7, respectively. $n_7 \equiv 1 \mod 7$ and $n_7 | 3 \implies n_7 = 1$ $n_3 \equiv 1 \mod 3$ and $n_3 | 7 \implies n_3 = 1, 7$ Let $P_3 \cong […]

Why is a finite group, with a maximal subgroup which is abelian, soluble?

I’ve come across an exercise saying the following: If $G$ is a finite group which contains a maximal subgroup $M$ which is abelian, show that $G$ is solvable and that $G^{(3)}$ (the third term in the derived series) equals 1. I can see that it’s sufficient to show that $G^{(2)}$ is a subgroup of $M$, […]

How many normal subgroups?

$(a)$ Let $T_1$, $T_2$, $\cdots$, $T_k$ be non-abelian finite simple groups. How many normal subgroups does the direct product $T_1 \times T_2 \times \cdots \times T_k$ have? $(b)$ Let $G$ denote the elementary abelian group of order $p^n$, i.e. $$G \cong \underbrace{(\mathbb{Z}/p\mathbb{Z}) \times \cdots \times (\mathbb{Z}/p\mathbb{Z})}_{n\text{ times}}$$ How many subgroups of order $p^k$ does $G$ […]

Subgroups of the Symmetric Group

So I am aware a subgroup $H \subset S_N$ can consist of only even permutations (i.e. taking the set of all even permutations will produce a normal subgroup), but can it consist of only odd permutations?

The Schur Theorem

The Schur Theorem: if $\left\vert G:Z\left( G\right) \right\vert $ is finite, then $G^{\prime }$ is finite. My question is: if $1\not=N\trianglelefteq G$ such that $\left\vert G:NZ\left( G\right) \right\vert $ is finite, is there some information on $G^{\prime }$ or some finiteness conditions involving $G^{\prime }?$