# 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.

49,514 questions
Filter by
Sorted by
Tagged with
4 views

### How to find different representation of permutation in product of transpositions?

It is known that product of transposition to write is not unique. it can be written in many ways,there is theorem that stats that all representation of product of transposition for a given cycle are ...
24 views

• 122
36 views

### $G = \langle C_G(a) \mid a \in Q \setminus \{1\} \rangle.$ Proof by induction on $|G|$

I have a question on a proof found in our lecture book on group theory from Gernot Stroth. I do not understand what here is meant by "Induction on $|G|$", which makes it very hard to grasp ...
1 vote
60 views

### Explicitly finding a finite abelian group from a presentation, $\langle a, b~|~a^n = b^n=1, ab=ba, ab^{-1}ab^{-1}=1\rangle$

I have the following group presentation: $G= \langle a, b~|~a^n = b^n=1, ab=ba, ab^{-1}ab^{-1}=1\rangle$ It is clear that $G$ is a finite abelian group. I am interested in knowing what exactly $G$ ...
• 127
42 views

### Every solvable group of order less than 200 must be of order 60,120,168 or 180

I am trying to prove that every solvable group with order less than 200 must have order 60,120,168 or 180. I thought of taking Burnside’s Theorem and rejecting everyone of the groups of order $p^a q^b$...
1 vote
65 views

• 149
35 views

### Consider the function $\phi(x)=\ln x$ from $\mathbb R^+$ (under mult) to the group $\mathbb R$ (under add) . Prove that $\phi$ is an isomorphism. [duplicate]

Okay so I've proved all of the parts needed for an isomorphism except to show that $\phi(x)$ is onto. How do I go about doing that? I know that you start by assuming that $y \in \mathbb{R^+}$ and that ...
65 views

• 914
67 views

• 3
1 vote
32 views

### permutation of the determinant according to the groups $S_{n1}$ and $S_{n2}$?

Reading about alternating linear n maps, I found this alternative definition of a determinant, based on its permutation expression (which iteratively sums the product of all the permutations that can ...
• 31
168 views

### Does this type of short exact sequence always split?

Consider the family of short exact sequences $$0 \to \mathbb{Z}^m \to G \to \mathbb{Z}/n \mathbb{Z} \to 0$$ where $G$ is a finitely generated abelian group. This is known to not always split, as per ...
• 2,037
1 vote
89 views

### $g$ lies in the centre of $G$ iff $\vert\chi(g)\vert=\vert\chi(1)\vert$ for all irreducible characters $\chi$

I have been trying the following exercise (Chapter 3, Exercise 10) from Webb's "A Course in Finite Group Representation Theory". Let $g\in G$. Show that $g$ is in the centre of $G$ if and ...
• 15
36 views

### isomorphic groups that are G-equivariant [closed]

Start with two finite groups $A$ and $B$ and a group isomorphism $f$ between them. Let a finite group $G$ act on both $A$ and $B$. By definition $f$ is $G$-equivariant if $g(f(a))=f(g(a))$. Do I ...
• 3
18 views

• 271
1 vote
53 views

### How to show the following sequence of group homomorphism to be exact?

Background The following is taken from: Graduate Course In Algebra, A - Volume 1 by: Ioannis Farmakis and Martin Moskowitz ....a short exact sequence of groups is just another way of talking about a ...
• 3,429
1 vote
28 views

### How to show the following sequence of group homomorphism is exact?

Background The following is taken from: Graduate Course In Algebra, A - Volume 1 by: Ioannis Farmakis and Martin Moskowitz ....a short exact sequence of groups is just another way of talking about a ...
• 3,429
51 views

### $G = \mathbf{Z}\times\mathbf{Z}$ and its normal subgroup $\langle (1,0)\rangle$. How is $G/N$ isomorphic to $\mathbf{Z}$? [duplicate]

I am new to quotient spaces and I came across this problem and it confused me: Given the group $G=\mathbf{Z}\times\mathbf{Z}$ and some normal subgroup $\langle (1,0)\rangle$.. The quotient group $G/N$ ...
1 vote
25 views

### Constructing compatible functions $\phi'_i,\phi'_j$ for equivalence relation $\sim$ for directed limit of directed systems of groups.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz, and How to Prove it by: Dan Velleman. Definition 1: Let $(G_i)_{i\in I}$ be ...
• 3,429
### Let $H=\{A\in G\; |\; \text{det}(A)=3^k, k\in\mathbb Z\}$, $G=GL(2,\mathbb R)$. $H\le G$ . Prove that $H$ is a normal subgroup of $GL(2,\mathbb R)$ [closed]
Not really sure how to solve this one. The full question is: Let $H=\{A\in G\; |\; \text{det}(A)=3^k, k\in\mathbb Z\}$, where $G=GL(2,\mathbb R)$. It is a fact that $H\le G$ (and you don't need to ...