# Questions tagged [group-theory]

For questions about the study of algebraic structures consisting of a set of elements together with a well-defined binary operation that satisfies three conditions: associativity, identity and invertibility.

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### Why is studying centralizers the/a key to classifying finite groups?

In this MO thread https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable, Borcherds says One problem, as least with the current methods of ...
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### Number of involutions in $S_n$? [duplicate]

I was having some fun with the number of involutions (didn't know they were called that) in the symmetric group $\Psi(S_n$), and tried to come up with a simple formula for it, I'm like, so close to ...
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### Which wallpaper groups do these images belong to?

The three images below are 'pieces' of infinite wallpaper patterns. The first one corresponds to an infinite checkerboard pattern with alternating white and black squares. The second image corresponds ...
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### Conditions for a representation of a group. (updated)

My lecturer wrote that a matrix representation of a group $G$ is a group homomorphism $\rho$ from $G$ to the set of invertible square matrices over a field. He proceeded to write that this is ...
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### Using properties of bijection for quicker proof of conjugate subgroup symmetry

My textbook, Szekeres's "A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry", tacitly alludes to the following lemma, which I've tried to write out and ...
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### Axioms for finite cyclic groups

tl;dr: What is a minimal definition of a finite cyclic group, without resorting to the definition of a group? Let $G$ be a finite set and $\star$ be a binary operation on $G$. My question is whether a ...
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### show that Ĝ is non empty and an abelian group when equipped with pointwise operations. [closed]

Assuming G is a locally compact group ( not abelian). A character of G is a continuous group homomorphism x: G → 𝕋 , where 𝕋 is the circle group . Denote by Ĝ the set of characters of G. Here how ...
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### Definition of Presentation of Group in Dummit’s Abstract Algebra [closed]

A subset $S$ of elements of a group $G$ with the property that every element of $G$ can be written as a (finite) product of elements of $S$ and their inverses is called a set of generators of $G$. We ...
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### On coset equalities

Let $H \le \textrm{Sym}(n)$, where $n \in \mathbb{Z}$ and Sym(n) stands for the symmetric group over $n$ elements. Furthermore let $g \in \textrm{Sym}(m)$ where $m \in \mathbb{Z} \land m \ge n$. This -...
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### Why is Cayley's theorem stated for finite groups?

My textbook writes Cayley's theorem on groups as follows: Every finite group $G$ of order $n$ is isomorphic to a permutation group. By permutation group is meant a subgroup of the group of ...
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### Prove: $\mathbb{C}^* \cong U_1 \times \mathbb{R}^*$ (isomorph to circle group)

Define the circle group $U_1 = \{z \in \mathbb{C} \mid |z| =1 \} \subset \mathbb{C}^*$. Prove: $\mathbb{C}^* \cong U_1 \times \mathbb{R}_{>0}^*$. I would like to know if I did it correct. I ...
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### Is this proof that a homomorphism preserves identities correct (sufficient)?

My textbook writes the following proof for the statement in the title: For a homomorphism $\varphi:G \to G'$ and any $g\in G$, $\varphi(g) = \varphi(ge) = \varphi(g)\varphi(e)$. Multiplying both ...
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### Is there a name for a group whose elements are Hurwitz?

I am not familiar much with group theory except for some basics. I know that a group is defined by some axioms for addition and multiplications. I have heard of general linear group where invertible ...
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### $\phi:G\to \bar G$ is an isomorphism. Then the equation $x^k=b$ has the same number of solutions in $G$ as the equation $x^k=\phi(b)$ in $\bar G$

This is my first post here, so forgive for any formatting errors or lack of information; please let me know if I can rectify any issue with my post. I've been stuck on this problem for quite a bit, ...
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### Moebius transformation on circles

I'm working on a program that draws the limit sets of Schottky groups defined by pairs of circles. If circle $C$ with radius $r$ is centered at $P$, then to map that circle to some other circle $C'$ ...
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### Sylow 2-subgroup of $GL(2,\mathbb{Z}_{2^n})$ [closed]

The structure of Sylow 2-subgroups of $GL(2,q)$ (or even of $GL(n,q)$) is well known, but I couldn't find any information about Sylow 2-subgroup of $GL(2,\mathbb{Z}_{2^n})$, so could anyone help me to ...
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### Defining $G$-module from representation.

Consider a group $G$ and a representation $\rho=(\rho_{ij})$ of degree $d$ as well as a $d$-dimensional vector space $V$. In the book Algebra by P. M. Cohn he states that we can turn $V$ into a $G$-...
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