# Questions tagged [group-theory]

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 is the study of groups.

33,537 questions
35 views

### Structure of сonjugacy subgroup intersection graph

Let $G$ be a finite group, $H$ is proper subgroup and ${\cal H}(H)$ the set of all subgroups of conjugate H. Construct the graph $\Gamma$, with vertices ${\cal H}(H)$ and two subgroups adjacent if ...
12 views

34 views

### Is addition on a circle equivalent to addition of integers [closed]

If you do modulo arithmetic on a circle you have a certain group. But if you fourier transform a function on a circle you get discrete values. (Compared with continuous fourier tranform of a function ...
49 views

### Klein four groups, cyclic groups

A basic definition question: "the Klein four group V is the simplest group that is not cyclic." Does this simply mean you need two (non-identity) elements of the group to generate the entire group? ...
43 views

### Visualization of the relationship between Conjugacy Classes and Centraliser

I've recently come across a theorem in my abstract algebra course which states that for a finite group $G$ and for any $a\in G$ $|\text{cl}(a)|=[G:C(a)]$ where $\text{cl}(a)$ is the conjugacy class ...
501 views

### Can all groups be thought of as the symmetries of a geometrical object?

It is often said that we can think of groups as the symmetries of some mathematical object. Usual examples involve geometrical objects, for instance we can think of $\mathbb{S}_3$ as the collection of ...
76 views

### Uniqueness of Rank of Free Group via Abstract Nonsense

It is a well known result that given two isomorphic free groups, their freely generating sets have the same cardinality, which is then called the rank of a free group. I am well aware of the usual ...
18 views

80 views

### Every subgroup of cyclic group is cyclic,How do I visualize this fact graphically? [closed]

I want to visualize everything in Mathematics.Is there a way to visualize the theorem I stated above?Can it be represented graphically?I want to be very analytical in this topic.So please can anyone ...
64 views
+50

94 views

### On finite simple groups whose orders are perfect powers

A short note in Group Atlas v2.0 states: $\mathrm{PSp}(4,7)$ is the smallest simple group whose order is a proper power. Question: Is there other known finite simple groups whose orders are ...
73 views

### how many abelian transitive subgroups of $S_{n}$

It's well-known that any abelian transitive subgroup A of a symmetric group $S_{n}$ has order $n$. Moreover, does anyone know how many abelian transitive subgroups of $S_{n}$ and what do they look ...