Articles of group theory

Is every finitely generated reflection group a Coxeter group?

Suppose $G$ is a subgroup of $O_n(\mathbb{R})$ generated by finitely many reflections, where a reflection is defined to be a linear transformation fixing a hyperplane and sending a normal vector to the hyperplane to its negative. Suppose the normal vectors corresponding to these hyperplanes are linearly independent. Is G necessarily isomorphic to a Coxeter group? […]

Is a monoid finitely generated by finite order elements periodic?

This may be a simple question, but I can’t make it out right now. Let $M$ be a monoid, and $g_1, \ldots, g_n \in M$ be elements of finite order in $M$. Is $\langle g_1, \ldots, g_n \rangle$ periodic (i.e, have all its elements of finite order)? The monoid I’m interested in is the multiplicative […]

Prove that G is a group

The exercise is: Let $g\in G$. $G$ is a group. Prove that $G=\{gx:x\in G$}. I know the the definition of group but the proof that is in the book is the next one: Let $H=\{gx:x\in G\}$ $H\subseteq G$ because $g\in G$ and $x\in G$ $\Rightarrow gx \in G$ G $\subseteq H$ (?) (by definition) Let […]

Show that $\det \rho(g)=1$ for elements g of odd order and $\det \rho(g)=-1$ for elements g of even order.

$G$ is a finite group. $\rho$ is a representation of $G$, then $\rho \mapsto \rho(g)$ is a $1$-dimensional representation. Show that if $\det \rho(g)=-1$ for some $g \in G$ then $G$ has a normal subgroup of index $2$. I believe this is because $$1=(-1)^2=(\det \rho(g))^2=\det \rho(g)\det \rho(g)=\det \rho(g^2)$$ so $g \in \ker(\det \rho)$, $\det \rho$ […]

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of $F$ have a square root. (Hint: For some integer $i$, ($1\leq i \leq s-1$), $i$ is odd if and only if $(s-1)+i$ is even.) My attempt: If $s=2^m$, then $s$ is even. This means $s-1$ is […]

Characteristics subgroups, normal subgroups and the commutator.

Let $G$ be a group and let $N$ be a normal subgroup of $G.$ Let $N’$ denote the commutator of $N.$ Prove that $N’$ is a normal subgroup of $G.$ What I do know is that the commutator subgroup is characteristics. What I am not sure about is whether or not $N’$ is characteristic in […]

Group of order 4k+2. Prove that the following permutation is odd.

Let $G$ be a group such that $|G|=4k+2$, and let $a \in G$ such that $a$ has order 2. Consider $f:G \rightarrow G$ given by $f(g)=ag$, $g\in G$. Prove that $f$ is an odd permutation. I was discussing the following problem with a friend, we sort of came up with the same answer, but we […]

A question about perfect group

Let $G$ be a finite group. Show that if $G = G’$, then $Z$$\left( G/{Z\left( G \right)} \right)=1$. My attemp is here. Fact: $G’ \le N$ if the quotient group $G/N$ is abelian. Since $G = G’$, by the fact, there is no normal subgroup $N$ of $G$ such that $G/N$ is abelian. Suppose that […]

under what conditions a product of matrices is the identity matrix (more complicated than that)?

I have a set of matrices $A_1,\ldots,A_n$ and another set $B_i = A_i^{-1}$ for $i = 1,\ldots, n$ (I assume $A_i$ are invertible). Let $\mathcal{A} = \{A_i\} \cup \{B_i\}$. What are some simple conditions under which $$\prod_{k=1}^r C_k = I$$ (for $C_k \in \mathcal{A}$ and $r$ an integer) if and only if the $C_k$ are […]

Sylow subgroups of a group G and product of subgroups

Let $G$ be a finite group. Let $H \leq K\unlhd G$. If for each $P$ Sylow subgroup of G there exists $x \in G$ such that $HP^{x}=P^{x}H$ then for each $P \cap K$ of $K$ there exists $y \in K$ such that $H(P\cap K)^{y}=(P\cap K)^{y}H$. I know that $H(P\cap K)^{x}=(P\cap K)^{x}H$, but $x$ may not […]