# 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.

40 views

### Is it possible for a group to be a finite union of subgroups of infinite index?

Just restating the title: Does there exist a group $G$ and subgroups $H_1, \ldots, H_k$ so that $[G:H_i]$ is infinite for each $i = 1, 2, \ldots, k$, and $G = H_1 \cup \cdots \cup H_k$? If $G$ is a ...
31 views

### Let $G$ be a group and show that $|G:Z(G)|$ cannot be prime.

Let $G$ be a group and show that $|G:Z(G)|$ cannot be prime where $Z(G)$ is the center of $G$ and $|G:Z(G)|$ is the index of $Z(G)$ in $G$. This was in a test that I had recently but I was not able ...
36 views

34 views

### If $p^n \mid |G|$ then the finite p-group $G$ has a normal subgroup of order $p^n$

Suppose that $G$ is a finite p-group. Show that if $p^n \mid |G|$ then $G$ has a normal subgroup of order $p^n$. I am stuck on this problem. I know that $G$ has subgroups of order $p^m$ but I don't ...
47 views

### Group theory puzzle (whirligig?)

This is a problem inspired from the Tomb Raider Anniversary video game (2007?) at the beginning of the Sanctuary of Scion in Egypt when Lara needed to solve the puzzle with 4 columns where the columns ...
46 views

55 views

### Outer Automorphisms of PSL2(R)

As far as I've been able to tell, a description of Out$(PSL_2(\mathbb{R}))$ isn't available online. I also looked in Lang's $SL_2(\mathbb{R})$ but it's not discussed. I guess my first question would ...
101 views

### non-abelian groups of order $p^2q^2$.

Let $p<q$ be prime numbers and let $G$ be a group of order $p^2q^2$. I wish to determine up to isomorphism how many groups $G$ are there. What I know: The abelian case is very clear. Moreover, ...
31 views

### Why is this proof incorrect? (Group Theory)

If $G$ is a finite non-trivial group and $H \leq G$ has index $p$ prime, then $H \lhd G$. This statement is actually false. However, I cannot find the error in what I wrote. Let $G$ act on the ...
39 views

### Proving that the multiplicative group modulo $2^r$ are cyclic iff $r<3$

No idea how to start, can someone give me a hint? Show that $(\mathbb{Z}/2^r\mathbb{Z})^*$ is a cyclic group if and only if $r<3$.
41 views

### $k$ cosets of normal subgroup and $k$th power of group element

I wrote a proof, but I'm afraid it's not quite perfect. If one wants to prove that for a finite group $G$, where $N \trianglelefteq G$, and $|G:N|=k$, then $a^k \in N$ for all $a \in G$, he can ...
39 views

167 views

### Group of integer orthogonal matrices

Let $O_n(\mathbb Z)$ be a group of orthogonal matrices B st. $B*B^T=I$ with entries $b_{ij} \in \mathbb Z$. How do I show that $O_n(\mathbb Z)$ is a finite group and find its order. I need to show ...
46 views

### Sufficient condition to check whether a group is Noetherian

Suppose $G$ is a finitely generated group. What conditions on $G$(or some subgroups of $G$) will force it to be a Noetherian group? Of course, if all subgroups are finitely generated or ACC on ...
Assuming that given operation is commutative and associative, are following definitions equivalent? 1) A zero element $e_0$ exists such that $e_i+e_0=e_i$ and inverse $e_{i'}$ exists for every $e_i$ ...