Articles of group theory

Existence of intermediate subgroups and representations theory.

Let $G$ be a finite group, $V$ an irreducible representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Suppose that $dim(V^H)>1$. Then $dim(V) > 1 $ and $dim(V^G)=0$. Question: Is there an intermediate subgroup $H \subset P \subset G$ such that $dim(V^P)=1$ ?

$X$ a set and $G$ a group. Let $G^X$ be the set of mappings from $X$ to $G$, show that $G^X$ can have same structure.

So I’ve got this exercise: Let $X$ be a set and $(G, \star)$ a group. We denote $G^X$ the set of a mappings from $X$ to $G$. Show that $G^X$ has a group structure induced by $G$ So this is my attempt. I first have to show that it has the identity element, but I […]

Cancellation property in groups

This question already has an answer here: How to prove $b=c$ if $ab=ac$ (cancellation law in groups)? 6 answers

Understanding the quotient of infinite groups $\mathbb{R}^2/H$ where $H = \{(a, 0): a\in \mathbb{R}\}$

Define $H = \{(a, 0): a\in \mathbb{R}\}$. Without using the fundamental homomorphism theorem, how would we know what $\mathbb{R}^2/H$ is? The quotient group is $\{H, (x, y) + H, (x_2, y_2) + H, \dots \}$. Intuitively, each coset in the quotient group is a horizontal line crossing through the point $(x_i + a, y_i)$ or […]

Necessary and Sufficient Condition for $\phi(i) = g^i$ as a homomorphism – Fraleigh p. 135 13.55

Let $g \in \text{ group } G $ and $n \in N$. Let $\phi : \mathbb{Z_n} \rightarrow G$ be defined by $\phi(i) = g^i$ for $0 \le i \le n$. Give a necessary and sufficient condition (in terms of g and n) for $\phi$ to be a homomorphism. Prove your assertion. My $g =$ solution’s […]

A second isomorphism theorem for action on cosets

Let $G$ be a finite group, and $K$, $L$ subgroups of $G$ such that $G = KL=LK$. Let $\Omega = G/K$ and $\pi: G \to S_{\Omega}$ the canonical action on cosets. Question: Is it true that $\forall g \in G$ $\exists l \in L $ such that $\pi(g)=\pi(l)$ ? If yes, then $\forall k \in […]

$S_n$ 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