Articles of group theory

Minimal sets of generators for groups

Does any infinite group contain a minimal set of generators? This is not true for semigroups. But for groups?

Homomorphism and normal subgroups

Suppose that $\phi : G \to G’$ is a homomorphism between the groups $G$ and $G’$. Let $N’$ be a normal subgroup of $G’.$ Prove that the inverse image of $N’$ is a normal subgroup of $G$. How can I prove this using the defintions? A proof that was given confused me. The proof states: […]

How are composition series for two isomorphic groups related?

If $G$ and $H$ are groups, $G\cong H$, and $$G=N_{0}\geq N_1\geq \dots \geq N_s ={e_G}$$ and $$H=M_{0}\geq M_1\geq \dots \geq M_t ={e_H}$$ are composition series for $G$ and $H$, what can be said about the relationship between the $N_i$s and the $M_i$s? By Jordan-Holder, we know that any two composition series for a particular group […]

What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?

What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters? I believe the answer for this is 6. As we can write the group elements as below (a)(b)(C) (ab)(c) (ac)(b) (bc)(a) (abc) (bac) Can we generalize that for any $S_n$ onto $n$ the number of automorphisms will be $n!$ Also […]

Proving the direct product D of two groups G & H has a normal subgroup N such that N isomorphic to G and D/N isomorphic to H

Let $D = G \times H$ be the direct product of groups $G$ and $H$. Prove that $D$ has a normal subgroup $N$, such that $N$ is isomorphic to $G$ and $D/N$ is isomorphic to $H$. Here’s where I stand…I know what a direct product is and I know what a normal subgroup is but […]

Proof on a particular property of cyclic groups

There is a theorem that states: $$|a^k| =\frac{n}{\gcd (n,k)}$$ The author begin by stating we are to prove that if d is a positive divisor of n, then $$|a^d|=\frac{n}{d}$$ Where did this came about? Could someone fill me in on the leap in argument?

Generalized Quaternion Group

Let $w = e^{\Large\frac{i\pi}{n}} \in \mathbb{C}.$ Prove that the matrices $X=\left( \begin{array}{cc} w & 0 \\ 0 & \overline{w} \\ \end{array} \right)$ and $Y = \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array} \right)$ generate a subgroup $Q_{2n}$ of order $4n$ in $\operatorname{GL}(2, \mathbb{C})$, with presentation $\langle x,y \mid x^n=y^2, x^{2n}=1, y^{-1}xy=x^{-1}\rangle.$

A subgroup of a cyclic normal subgroup of a Group is Normal

Is it true that Subgroup of a Cyclic Normal subgroup of a Group is again Normal ? If so any hints for the proof?

Action of a group on itself by conjugation is faithful $\iff$ trivial center

Definition 2.15. A group action of $G$ on $X$ is called faithful (or effective) if different elements of $G$ act on $X$ in different ways: when $g_1\neq g_2$ in G, there is an $x\in X$ such that $g_1\cdot x \neq g_2\cdot x$. Example 2.17. The action of $G$ on itself by conjugation is faithful if […]

What is a conjugacy class of reflection?

I have a problem to do, asking to show that $D_{2n}$ has two conjugacy classes of reflections if n is even, but only one if n is odd. My question is, what is a conjugacy class of reflection? I have seen that there is a solution to this question on this site, but I don’t […]