Articles of group presentation

Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: […] find a set of generators of the first group that satisfies the relations of the second group […] But I doubt that, as this just shows that the first group is an epimorhpic image of the second group. For example […]

Generalized Quaternion Group

Let $w = e^{\Large\frac{i\pi}{n}} \in \mathbb{C}.$ Prove that the matrices $X=\left( \begin{array}{cc} w & 0 \\ 0 & \overline{w} \\ \end{array} \right)$ and $Y = \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array} \right)$ generate a subgroup $Q_{2n}$ of order $4n$ in $\operatorname{GL}(2, \mathbb{C})$, with presentation $\langle x,y \mid x^n=y^2, x^{2n}=1, y^{-1}xy=x^{-1}\rangle.$

Determining the presentation matrix for a module

I am trying to study some module theory using the book “Algebra” by Michael Artin (2nd Edition, to be precise), and I can’t really fathom what is written in Section 14.5. Left multiplication by an $m \times n$ matrix defines a homomorphism of $R$-modules $A: R^n \rightarrow R^m$. Its image consists of all linear combinations […]

Presentations of Semidirect Product of Groups

I have seen here that given two groups $G=\langle X|R \rangle := F(X)/N(R)$ and $H=\langle Y|S \rangle := F(Y)/N(S)$, then their semidirect product can be written as: $$ G\rtimes_\phi H \;=\; \langle X, Y \mid R,\,S,\,yxy^{-1}=\phi(y)(x)\text{ for all }x\in X\text{ and }y\in Y\rangle \tag{1} $$ I’m trying to prove this result with an epimorfism $\theta:F(G\times […]

Orders of Elements in Minimal Generating sets of Abelian p-Groups

I’m looking for as much information about the orders of elements in minimal generating sets of finite abelian $p$-groups as possible. What I really need is complete knowledge about the possible orders of elements in such groups and how many of each order there can be. I know that every group of the form $(\mathbb […]

Understanding group presentation as a quotient

I’m just starting to learn a little group theory, so please forgive any ignorance I demonstrate in the following. I’m trying to understand the concept of a group being defined based on its presentation $G = \langle S \mid R \rangle$ as the quotient group of the free group $F_S$ generated by $S$, and the […]

How to show this presentation of the additive group $(\mathbb{Q},+)$?

The task is: Show that $$ \langle (x_n)_{ n \in \mathbb{N}} \mid x_n^n = x_{n-1} \text{ for } 1 < n \in \mathbb{N} \rangle $$ is presentation of additive group $(\mathbb{Q},+)$. Can you explain me how to show this? Or explain simple what is presentation?

Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$…

Looking at the images below, you recognize that the adajency matrix of the graph $A_G$ splits up into three different color submatrices, with $A_G=A_r+A_b+A_d$ (where $d$ is dark, damn…). It’s obvious that $A_k^2=1$. Now have a look a the right coloring: You’ll see that $(A_dA_r)^2=(A_rA_b)^2=(A_bA_d)^2=1$ as well. Let’s use $a,b,c$ instead. We summarize this as: […]

groups of order $p^4$

I need the classification of finite non-abelian groups of order $p^4$ from E. Schenkman’s book “Group theory, 1965”. Unfortunately our library has no this book and there does not exist the full page version of the book on! It will be appreciated if someone who has the book, upload the related pages of the […]

Group presentations: What's in the kernel of $\phi$?

I have a question about group presentations (in terms of generators and relations). It’s been really bugging me for ages. Would really appreciate any thoughts on this. Cheers, Michael You are ‘given’ a group $G$. First, we find a list of generators for the group. Let’s be lazy and take all of the elements as […]