Articles of group theory

Classification of the decomposable primitive permutation groups

It is seen in comments here that the diagonal subgroup of the finite group $G \times G$ is core-free maximal iff $G$ is a nonabelian simple group. This gives examples of decomposable primitive permutation groups. Are there others examples? Question: What’s the classification of the decomposable primitive permutation groups?

Proofs that: $\text{Sp}(2n,\mathbb{C})$ is Lie Group and $\text{sp}(2n,\mathbb{C})$ is Lie Algebra

Consider following Lie Group: $$ \text{Sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}(\mathbb{C})\mid J=g^TJg\}\quad\ where\quad J=\begin{pmatrix} 0 & 1_n \\ -1_n & 0 \end{pmatrix} $$ And the corresponding Lie Algebra: $$ \text{sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}(\mathbb{C})\mid g^TJ+Jg=0\} $$ Are there any basic proofs that $\text{Sp}(2n,\mathbb{C})$ is a Lie Group and that $\text{sp}(2n,\mathbb{C})$ is the corresponding Lie Algebra without using submersions (seen here: Why is $Sp(2m)$ as […]

Matrix linear algebra generators

Linear algebra and special-linear group experts please help: It is known that in principle one can generate this $C$ matrix form the $A$ and $B$ matrix below. Here $$ C=\begin{pmatrix} 0& -1& 0\\ 1& 0& 0\\ 0& 0& 1 \end{pmatrix} $$ from: $$ A=\begin{pmatrix} 0& 0& 1\\ 1& 0& 0\\ 0& 1& 0 \end{pmatrix}, \text{ […]

Group theory – prove that $\forall x((x^{-1})^{-1}=x)$

so I got this question for homework: Prove that this property can be deduced from group theory: The inverse of an inverse is the identity: $\forall x((x^{-1})^{-1}=x)$ I tried building this statement from the group’s axioms but was having difficulties. Some help/hints? Thanks

How to find the quotient group $Z_{1023}^*/\langle 2\rangle$?

When $m=1023$, what are the quotient groups below? $$Z_m^*/ \langle 2\rangle$$ $Z_m^*=\{1,2,4,5, \dots \}$ $\langle 2\rangle=\{1,2,4,8,16,32,64,128,256,512\}$ $$\begin{align*} Z_m^*/\langle 2\rangle &=\{1,2,4,8,16,32,64,128,256,512\},\\ {} & \mathrel{\hphantom{=}}\{2,3,5,9,17,33,65,129,257,513\}, &&\text{(added by 1)},\\ {} & \mathrel{\hphantom{=}}\{3,4,6,10,18,34,66,130,258,514\}, && \text{(added by 2)},\\ {} & \mathrel{\hphantom{=}}\{5,6,8,12,20,36,68,132,260,516\}, &&\text{(added by 4)}\\ & \,\, \vdots \end{align*}$$ Are the answers $\langle 2 \rangle$ incremented by all the elements i$\in […]

If $F(a, b)=\langle a, B\rangle$ then $B=a^ib^{\epsilon}a^j$: a neat proof?

If you take a generating pair for $F(a, b)$, $(a, B)$, then it is intuitively obvious that $B=a^ib^{\epsilon}a^j$ for $i, j\in \mathbb{Z}$, $\epsilon=\pm 1$. However, I cannot come up with a neat proof of this, and so was wondering if someone could either provide a reference or come up with a more elegant approach. My […]

Prove that G is a cyclic group

Suppose that $|G| = pq$ where $p$ and $q$ are primes such that $p < q$ and $p$ does not divide $q − 1$. Prove that $G$ is a cyclic group. A cyclic group is a group that has a unique generator element, so is the way to go with this to find that element? […]

If $G$ is isomorphic to all non-trivial cyclic subgroups, prove that $G\cong \mathbb{Z}$ or $G\cong \mathbb{Z}_p$

Assuming $G$ is isomorphic to all its non-trivial subgroups, prove that $G$ is isomoprhic to either $\mathbb Z$ or $\mathbb Z_p$ for $p$ prime.

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

This question already has an answer here: Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$ 1 answer

Suppose that half of the elements of G have order 2 and the other half form a subgroup H of order n. Prove that H is an abelian subgroup of G.

Let $n>1$ be a positive integer. Let $G$ be a group of order $2n$. Suppose that half of the elements of G have order 2 and the other half forma a subgroup H of oder n. Prove that H is an abelian subgroup of G. I can only deduce from the index of H that […]