Articles of group theory

the representation of a free group

A group $G$ is generated by $\begin{pmatrix}1&n\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\n&1\end{pmatrix}$, then we know $G\cong \mathbb{F}_2$ which is a free group generated by two elements. Now I consider the representation: $G\to GL(2,\mathbb{R})$, it is necessary that the image of $\begin{pmatrix}1&n\\0&1\end{pmatrix}$ is a triangular matrix under conjugation? Thanks in advance.

Let $G=GL(2,\mathbb{Z}/5\mathbb{Z})$, the general linear group of $2 \times 2 $ matrices with entries from $\mathbb{Z}/5\mathbb{Z}$

Let $G=GL(2,\mathbb{Z}/5\mathbb{Z})$, the general linear group of $2 \times 2 $ matrices with entries from $\mathbb{Z}/5\mathbb{Z}$. i) List the elements of the centre of $G$. ii) Compute the order of $G$. iii) Give an example of an element of $G$ of order $4$, and of an element of $G$ of order $8$. iv) Let $x$ […]

Number of ways a group element of a finite group can be written as a given word

I had previously asked about the number of ways a group element in a finite group could be written as a commutator (the question is still open for a proof, by the way) In how many ways can a group element in a finite group be written as a commutator? Let $G$ be a finite […]

Simple proof for finite groups that $g^{\#(G)}=1$

Possible Duplicate: Is Lagrange's theorem the most basic result in finite group theory? I can’t seem to find a simple proof of this in my textbook, not can I figure out a good way to search for it online. Basically, I’m trying to show that $\forall g \in G$, $g^{\#(G)}=1$. Obviously I can show this […]

Condition to be a group.

Let $G$ be a semigroup. I’m showing that $G-group \iff [ \ \exists_{e\in G} \forall_{a\in G}: ea=a\ ] $ and $ [\ \forall_{a\in G}\exists_{a^{-1}\in G}: a^{-1}a=e \ ] $ “$\Rightarrow$” is obvious. “$\Leftarrow$” This is how I do it: Let $a \in G$. $ea=a$ $aea=aa$ $(ae)a=aa$ Is it true that this implies $ae=a$? How’s that […]

Is the center of a p-group non-trivial?

Is it true that if $G$ is a $p$-group, where $p$ is a prime, then center of $G$ is non-trivial? I know for finite $p$-group center of $G$ is non-trivial (easy to prove using class equation for group). But I am not sure about infinite $p$-group. I have no idea how to approach this problem. […]

Why is $xyzx^{-1}y^{-1}z^{-1}$ a commutator?

Exercise 1.6.41(a) of Bourbaki’s Algebra goes like this: Let $x,y$ be two elements of a group $G$. For there to exist $a,b$ in $G$ such that $bay=xab$, it is necessary and sufficient that $xy^{-1}$ be a commutator. The necessity part is nothing else than the statement that for arbitrary elements $x,a,b\in G$, $bayb^{-1}a^{-1}y^{-1}$ is a […]

How is this subgroup a normal subgroup?

Let $G$ be a group; let $N$ be a normal subgroup of $G$; and let $M$ be a characteristic subgroup of $N$. Then how to show that $M$ is normal in $G$? My work: Let $m \in M$, $g \in G$. We need to show that $gmg^{-1} \in M$. Now let $T$ be an automorphism […]

Fastest way to show that $D_6 \to S_5$ is an injective homomorphism

I want to show that there is an injective homomorphism from $D_6 \to S_5$ where $D_6$ denotes the dihidral group of order 12 and $S_5$ the symmetric group. But I’m not sure how I can do this efficiently. I define $f: D_6 \to S_5$ by $f(\sigma) = (12)$ and $f(\rho) = (123)(45)$, with $\sigma$ being […]

Maximal finite order of Abelian Groups

Let $G$ be a commutative group, and let $g$ be an element of $G$ be an element of maximal finite order, then $|h|\leq |g|$. Prove that in fact if $h$ is finite order in $G$, then $|h|$ divides $|g|$. This is what I have: Proof by contradiction If $|h|$ is finite but doesn’t divide $|g|$, […]