Articles of group theory

Proof for a subgroup $H$ of the finite subgroup $G$ there are at least $|H|$ elements that are not in conjugates of $H$

A problem in Isaac’s finite group theory is to prove if $G$ is a finite group and $H$ is a proper subgroup of $G$ then there are at least $|H|$ elements that are not of the form $ghg^{-1}$ with $g\in G$ and $h\in $H. Thank you and regards.

Von Dyck's theorem (group theory)

Did anyone find a proof of this theorem? I can’t find it on the Internet. The theorem is : Let $X$ be a set and let $R$ be a set of reduced words on $X$. Assume that a group $G$ has the presentation $\langle X \mid R \rangle$. If $H$ is any group generated by […]

can a group with non-trivial center be isomorphic to a normal subgroup of its group of automorphisms?

i think (tho would be grateful for error-check) that the line of reasoning below suggests any group with trivial center is isomorphic to a normal subgroup of its automorphism group. question does the converse hold? i.e. if a group is isomorphic to a normal subgroup of its automorphism group must it have a trivial center? […]

Where is the “relation” here?

Looking at the definition of a monoid it says that: A monoid is a set that is closed under an associative binary operation and has an identity element $I \in S$ such that for all $a \in S$, $I a = a I =a$ But what does $I a$ mean here? I mean it’s just […]

What does this group permute?

In a question, I’m given a group $G=Q_1 \times Q_2 \times \cdots \times Q_t$, where each $Q_i$ is a generalized quaternion group, and told it is a transitive permutation group. But what set does the group permute?

Burnside's formula on a hexagon

I was trying to use Burnside’s formula on a regular hexagon. I believe the answer is $13$, but I am trying to show my work. Here is what I have figured out. $$\frac{1}{12}(2^6 + 36 + 30 + \cdots?)$$ $2^6$ is $D_{6}$ (at least that is what I think the regular hexagon is called). Then […]

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 […]