# Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

58 views

### How to find a onto homomorphism between two groups?

Consider the following subgroups of $\text{SL}(2,\mathbb{Z})$ : $A$ the subgroup of matrices with determinant $1$ : \begin{bmatrix}4\mathbb{Z}+1&8\mathbb{Z}\\4\mathbb{Z}&4\mathbb{Z}+1\end{...
42 views

### How to find out generators of the following free group?

The following is the quotient group of SL($2,\mathbb{Z}$). Consider $(H/\{-1,1\} \cap H)$ where $H=\begin{bmatrix}2\mathbb{Z}+1&4\mathbb{Z}\\2\mathbb{Z}&2\mathbb{Z}+1\end{bmatrix}$ How do ...
36 views

### how to find the index of following subgroup?

if I denotes the principal congurence group of level 2 i.e. $I=\{ M \in SL(2,Z) ; \:M \:\:\text{congruent to I} \mod(2)\}$. or I= \begin{bmatrix}2\mathbb{Z}+1&2\mathbb{Z}\\2\mathbb{Z}&2\...
42 views

30 views

### Let $X : S_3 → GL_2(\mathbb{R})$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group $S_3$...
22 views

### Several true/false statements about a finite group $a,g\in G$ such that $a$ is of order $2$

Let $G$ be a finite group, and $a,g\in G$ such that $a$ is of order $2$, then the following is either true or false: The element $gag^{-1}$ is of order $2$. $(ag)^2=g^2$ if $ag$ is of ...
64 views

30 views

### reference request Schur Zassenhaus Theorem

I am looking for a reference for the Schur Zassenhaus Theorem, saying that any normal Hall subgroup admits a complement. An on-line search show that it is supposed to be in "The theory of groups" by ...
36 views

164 views

### Only two groups of order $10$: $C_{10}$ and $D_{10}$

Show that up to isomorphism there exist only two groups of order 10: $C_{10}$ and $D_{10}$. I need some help on this question. I only know the basic definitions of isomorphism. Any hints?
124 views

### “conjugate to/with” or “conjugated to/with”, a terminology question in group theory.

This is a terminology question from a non-native English speaker. Let $G$ be a group and $a,b\in G$ such that there exists $c\in G$ verifying : $$b=cac^{-1}$$ I could say : the element $a$ is ...
If K and Q are both groups and $h:Q\rightarrow \text{Aut}(K)$ is a homomorphism then the group operation for the semidirect product $K\rtimes_hQ$ is: $$(k_1,q_1)*(k_2,q_2)=(k_1h(q_1)(k_2),q_1q_2)$$ ...
### How many homomorphisms are there $t: \mathbb{Z}_3 \times U_8 \to S_5$?
This was a question in our quiz today and no one in class knew how to answer it correctly or are not sure). How many homomorphisms are there $t: \mathbb{Z}_3 \times U_8 \to S_5$, where $U_8$ is ...