Articles of group theory

$S_n$ acting transitively on $\{1, 2, \dots, n\}$

I am reading Dummit and Foote, and in Section 4.1: Group Actions and Permutation Representations they give the following example of a group action: The symmetric group $G = S_n$ acts transitively in its usual action as permutations on $A = \{1, 2, \dots, n\}$. Note that the stabilizer in $G$ of any point $i$ […]

Let $G$ be a set with associative binary operation and a unit.

Let $G$ be a set with associative binary operation and a unit. Assume that for every $ g \in G$ there exists $ x \in G$ with $xg = 1$. Prove that $gx = 1$ is a consequence. That above is the question, and i think i have the answer however I need clarification on […]

Text on Group Theory and Graphs

A student and I are going to investigate the use of group theoretic techniques in graph theory. What are good texts in this area (introductory and otherwise)? We are particularly interested in studying automorphism groups of graphs, but a text with a broader view would also be welcome.

If the size of 2 subgroups of G are coprime then why is their intersection is trivial?

Let H and K be subgroups of G, with size p and q respectively, where p and q are coprime, how can we show that H intersect K is {e} where e is the identity element in G

Frattini Property of a subgroup

A subgroup $H$ of $G$ is said to satisfy the Frattini Property if for any intermediate subgroups $K$ and $L$ such that $H \leq K\unlhd L$ implies that $L \leq N_L(H)K$ A subgroup $H$ of $G$ satisfies the Frattini Property $\iff$ for any $x\in G$, there exists $y\in H^{\langle x\rangle}$ such that $H^y = H^x$ […]

Equivalence of tensor reps & tensor products of reps

Let a finite-dimensional vector space $V$ over $\mathbb R$ or $\mathbb C$ with dual $V^*$ and a group $G$ be given. Let $\rho:G\to\mathrm{GL}(V)$ be a representation, and let $T_kV$ and $V^{\otimes k}$ denote the vector space of $k$-tensors on $V^*$ and the $k$-fold tensor product of $V$ with itself respectively. The $k$-fold tensor product representation […]

How to determine the number of isomorphism types of groups of a given order?

if $G$ is a group whose order is $n$ can we determine the number of isomorphism types for this number or not ? for instance, if $n=4$ we have 2 types, $Z_4$ and $Z_2 \times Z_2$ ” Klein 4-group” for any number n, is a similar calculation possible ? in other words, let $P$ is […]

Is this a correct way to think about specific examples of groups using the category theory definition?

I’ll say now, before anything else, that I probably don’t know what I’m talking about. This is more me making a (hopefully) educated guess about a topic I’m not too familiar with. I recently started learning the basics of category theory – the absolute basics, as in what a category is as well as basic […]

Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.

I was trying to prove that a normal, abelian subgroup of $G$, $N$ is in the center of $G$ given that $|\operatorname{Aut}(N)|$ and $|G/N|$ are relatively prime. The official question: Let $N$ be an abelian normal subgroup of a finite group $G$. Assume that the orders $|G/N|$ and $|\operatorname{Aut}(N)|$ are relatively prime. Prove that $N$ […]

A group that has a finite number of subgroups is finite

This question already has an answer here: Finite number of subgroups $\Rightarrow$ finite group 2 answers