Articles of group theory

Proving a defined group $(G,*)$ is isomorphic to $(\mathbb{R},+)$

I am studying abstract algebra and I have this question: Let $G=${$a\in\mathbb{R}|-1<a<1$} Defined an operation $*$ in $G$ with $a*b=\frac{a+b}{1+ab}$ for all $a,b \in G$ Show that $(G,*)$ and $(\mathbb{R},+)$ are isomorphic. I know that to prove $(G,*)$ and $(\mathbb{R},+)$ are isomorphic, I have to show that 1. both $(G,*)$ and $(\mathbb{R},+)$ are groups. 2. […]

How can I prove that this group is isomorphic to a semidirect product?

Let $G=\langle x,y,z:xy=yx,zx=x^2z,zy=yz\rangle $, I think that this group is a semidirect product, my first attempt was to prove that this was isomorphic to $\mathbb{Z}^2\rtimes_\varphi\mathbb{Z}$, where $\varphi(1)((1,0))=(2,0)$ and $\varphi(1)((0,1))=(0,1)$. If $x=((1,0),0),y=((0,1),0),z=((0,0),1)$, then $x,y,z$ satisfies those relations. But I noticed that $\varphi(1)\not\in\mathrm{Aut}(\mathbb{Z}^2)$. How can I fix this? (actually I’m not sure that $G$ is a semidirect […]

Integral identity graphs — smallest example

From Paulus Graphs. “The (25,2)-, (25,4)-, and (26,10)-Paulus graphs have the apparently rather unusual property of being both integral graphs (or asymmetric) and identity graphs (a graph spectrum consisting entirely of integers).” I once quietly conjectured this was impossible, but I was proven wrong. Very wrong. Eric Weisstein was amused enough by my reaction that […]

How many elements are there in the group of invertible $2\times 2$ matrices over the field of seven elements?

How many elements are there in the group of invertible $2\times 2$ matrices over the field of seven elements? Sorry I have no idea so nothing to say? Any clue!

Name for Cayley graph of a semigroups?

I did Google search and can’t find a good answer. I thought I should ask experts here. Cayley graph is defined for groups. My question is: Is there a special name for the Cayley graph of semigroups?

The number of Sylow subgroups on $G$ with $|G|=pqr$

I’m doing a part of an exercise and I don’t know how to go on. Here it goes: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup $P$. Also suppose that $p \not\equiv 1$ (mod $r$), $p \not\equiv 1$ (mod […]

Prove that a group of order 3 must be cyclic.

Is my argument correct, if not what is wrong with it. If the group $G$ is not cyclic, this means that there is not element of order 3. So if $G=\{e, a, b\}$ then $|a|=2$ and $|b|=2$, a contradiction because by Lagrange’s Theorem element order divide group order but 2 does not divide 3. Hence […]

Software for generating Cayley graphs of $\mathbb Z_n$?

Does it exist any program (for linux) which can generate a nice Cayley graph of any $\mathbb Z_n$? (If it’s possible to create such a graph at all, that is.) (where perhaps $n ≤ 100$ or something like that)

Existence of group of order $p$ in group of order $pq$, $p>q$

This question is related to Question on groups of order $pq$, but is different. It references the same exercise, but an earlier part. The exercise is: A group of order $pq$, $p>q$, contains a subgroup of order $p$ and a subgroup of order $q$. The part I’m having trouble with is showing that assuming there […]

Which theorem did Poincaré prove?

Two related elementary facts in group theory are sometimes called Poincaré’s theorems. If $H\lneq G$ and $[G:H]<\infty$, then there is $N\leq H$, $N\lhd G$ such that $[G:N]<\infty$. The intersection of a finite number of subgroups of finite index is of finite index. Did he prove both? Could you please give me references to the paper(s) […]