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I faced to the following problem and could to verify the first part of it: Let $q\in\mathbb Q-\{0\}$ and let $v_q:(\mathbb Q,+)\to (\mathbb Q,+)$ is defined as $$v_q(t)=qt$$ Then proved that $v_q$ …

I am thinking on this problem: If $N\lhd H×K$ then either $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially. I assume; $N$ is not abelian so, there is $(n,n')$ and $(m,m')$ in $N$ …

There was a question here on the site that was asked: Let $G$ be a finitely generated abelian group. Then prove that it is not isomorphic to $G/N$, for every subgroup $N≠⟨1⟩$. Would the answer for a …

I am thinking on the following problem: If $|G|=12$ and there is no element of order $2$ in its center then $3$-Sylow subgroup of $G$ cannot be normal in $G$. I was told that to assume the $3$-S …

Let $G$ is a group and $H<G$ such that $|G-H|<\infty$. Prove that $|G|<\infty$. Truthfully, there is a hint for it: $H$ cannot be an infinite subgroup. It is clear if $|H|<\infty$, since $| …

I am working on some basic context in Group Theory. I face to the following problem: Let $G=GL(2,\Bbb Q)$ and $H\leq G$ be the subgroup of all diagonal invertible matrices. Find $$H_G=\cap_{g\in G …

As a definition, if for a group $(G$|$\Omega)$; the orders of $G_{\omega}$ ($\omega$ in $\Omega$) are equal to eachother, then $G$ is said to be a $1/2$ -transitive group . Any example for such these …

Reviewing the book "An Introduction to the Group Theory" By J.J.Rotman, I have found out that I have neglected the way $G'$, the commutator subgroup, becomes a subgroup of a given group $G$. Unfortun …

Let $G$ is a group and $\varphi: G\longrightarrow A$ be any group homomorphism wherein $A$ is abelian. Then $$\text{Hom}(G,A)\cong \text{Hom} \left(\frac{G}{G'},A\right)$$ What we have here is, f …

Let $G=(\mathbb Q,+)$ and $r_1=p_1/q_1, r_2=p_2/q_2\in G$. I want to prove that: $\langle r_1,r_2\rangle\subseteq \langle\frac{1}{q_1q_2}\rangle$ If $r_1,r_2,...,r_n\in G$ then $\langle r_1,r_2,... …

We know that: Theorem: if $G=\prod_{i=1}^n G_i$ be the direct product of groups $G_1,G_2,...,G_n$ then there exists normal subgroups $\bar{G_i}\cong G_i( i=1...n)$ of $G$ such that $$G=\bar{G_1}\b …

There is a theorem that states: "Every polycyclic group has a normal poly-infinite cyclic subgroup of finite index. " I just read the proof of it and honestly found some difficulties in it. The main …

There is a problem in Problems in Group Theory by J.D.Dixon : 2.51 If a permutation group $G$ contains a minimal normal subgroup $N$ which is both transitive and abelian, then $G$ is primitive. …

A group $G$, acting on a set $A$, is a permutation group such that it has an element of order $30$. Prove that order of $A$ is greater than $10$. Here, I supposed that, $|A|=6$ and wanted to make …

There is an exercise telling that every finite subset of group ($\mathbb Q$,+) or of group $\mathbf{Z}_{p^\infty}$ generates a cyclic group itself. For the first group if $X= \left\{\frac{p_{1}}{q_{1 …