# 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.

41 views

### What can we say about the order of the group which is generated by $\langle u,v\rangle$

If a group $G$ is generated by $\langle u,v\rangle$ and order of $u$ is less than or equal to $4$ and order of $v$ is less than or equal to $3$. Then what can we say about the order of the group? It ...
133 views

### Product of all elements in finite abelian group equals its own inverse

Let $G = \{e, a_1, a_2, \dots, a_n\}$ be a finite abelian group and let $S$ be the set of all the elements of $G$ which are not equal to their own inverse. The set $S$ can be divided up into pairs so ...
68 views

### Finding the inverse of an element of $S_n$ and it's order [duplicate]

I have two questions, 1) What are the ways to find the inverse of an element of $S_n$? 2) What are the ways to find the order of an element of $S_n$?
44 views

### Prove that $\langle S \rangle = G$

Let $S\subset G$ be a finite group such that $\#2S\gt\#G$ Prove that $\langle S \rangle = G$ My idea was to start by taking $a \in G$ and proving that $ax^{-1}$ is also in $G$ if $x$ is in $S$, but ...
55 views

### Isomorphic relation between dihedral groups

Theorem Let $G,H$ be abelian groups such that $Dih(G)\cong Dih(H)$ If $G$ is finitely generated, then $G\cong H$. I'm curious whether "finitely generated" hypothesis can be removed. If it'...
76 views

86 views

### If $f:G \to H$ is a homomorphism with kernel $N$ and $K$ is a subgroup of $G$ then prove that $f^{-1}(f(K))=KN$

If $f:G \to H$ is a homomorphism with kernel $N$ and $K$ is a subgroup of $G$ then prove that $f^{-1}(f(K))=KN$ Ok, so what I know from this: $G$ and $H$ are groups that must preserve operation. It ...
94 views

### Can this lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$. Let the lattice $\mathcal{L}$ as follows: ...
27 views

### is the group $(\mathbb{Z}_n \times \{-1,1\}, *) \simeq D_n$

is the group $(\mathbb{Z}_n \times \{-1,1\}, *) \simeq D_n$ $*: (a,b) * (c,d) = (a+_n(c \cdot_n b),bd)$ Showing $(\mathbb{Z}_n \times \{-1,1\}, *)$ was a group was a previous question but I have ...