Articles of finite groups

Multiplicative group of integers modulo n

Consider the abelian group $U_n=\{a\in \mathbb{Z}_n:(a,n)=1\}$. Is there a natural way to understand it as a subgroup of any other interpretation of the cyclic group of order $n$. For example, consider the group of $n$th roots of unity which is also isomorphic to $\mathbb{Z}_n$. Now in this case, if I define $U_n$ as $\{e^{\frac{2\pi ik}{n}}:(k,n)=1\}$ […]

About subsets of finite groups with $A^{-1}A=G$ or $AA^{-1}=G$?

Regarding to the problems Does $A^{-1}A=G$ imply that $AA^{-1}=G$? and Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$?, we are looking for some subsets $A$ of $G$ with $\lfloor \frac{|G|}{2}\rfloor-2 \leq |A|\leq \lfloor \frac{|G|}{2}\rfloor$ such that $A^{-1}A=G$ and $AA^{-1}\neq G$ or $AA^{-1}=G$ and $A^{-1}A\neq G$. Do such $G$ (non-abelian) and $A$ exist? We propose the […]

FC groups with infinite derived subgroup which are not constructed by direct product of finite groups.

A group $G$ is called FC (finite conjugacny) if every conjugacy class $C$ of $G$ has a finite order. It is called FD if the derived subgroup (constructed by commutators) is finite. It is clear that every FD group is also FC, but not vice versa. For example, the restricted (external) direct product of a […]

Every finite group of square-free order is soluble

prove that every finite group which it’s order is square free is soluble. I think it is enough to show that every sylow subgroup of this is cyclic. please tell me if my idea is right and if it is wrong please give me a little help,thanks.

The product of finitely many cyclic groups is cyclic

How to prove that the direct product of finitely many cyclic groups $C_{n_1}\times C_{n_2}\times\cdots\times C_{n_m}$ is cyclic if the $n_i$’s are pairwise relatively prime?

The Jordan-Hölder Theorem

I want to solve the following exercise from Dummit & Foote’s Abstract Algebra (exercise 10 in page 106): Prove part (2) of the Jordan-Hölder Theorem by induction on $\min\{r,s\}$. [Apply the inductive hypothesis to $H=N_{r-1} \cap M_{s-1}$ and use the preceding exercises.] The Jordan-Hölder Theorem as stated in the book is: Let $G$ be a […]

Subgroup of matrices exercise

Let $G=\{\left( \begin{array}{ccc}1 & b \\ 0 & a \\ \end{array} \right) : a,b \in \mathbb Z_7, a \neq 0\}$. Find the order of $G$. For each prime $p$ such that $p$ divides $|G|$, find all the elemens in $G$ that have order $p$. I know that $G$ is a subset of the $2 \times […]

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 […]

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 […]

Question about homomorphism of cyclic group

if $\varphi: G\to H$ is homomorphism. How do I prove that if $a\in G$ have finite order so $\varphi(a)$ had finite order to, and that:$$ord(\varphi(a))\mid ord(a)$$ Thank you!