# Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

37 views

### Subgroup of semidirect product

Let $G$ be a semidirect product of a normal subgroup $A$ with a subgroup $B$ and Let $H$ be a subgroup of $G$ such that $H\cap A$ is trivial. Is it true that $H$ is contained in a conjugate of $B$ ? ...
52 views

### Does there exist any infinite group $G$ whose each element is of order $2$?

Does there exist any infinite group $G$ whose each element is of order $2$? It is clear that the group should be abelian.I try to find out a suitable example to meet my purpose.But I have not yet ...
41 views

### Supersolvable groups and sylow towers

Is it true that a supesolvable group has a sylow tower? How can I construct the tower kwowing than there's a principal serier with factors of prime order? Can anyone help me with an hint or an idea? ...
24 views

### Finitely generated Ideals of finite algebras

i would really appreciate any help with this question. So, the question is: How to prove that finitely generated ideals of finite algebras over the ring F are finite over F? Thanks.
33 views

49 views

### Prove that a normal subgroup $K$ is characteristic in a finite group $G$ if $\gcd(|K|, |G/K|) = 1$

This is a part of an exercise $2.2$ on a page $203$ in a book "Algebra: Chapter 0" by P.Aluffi. Let $K$ be a normal subgroup of a finite group $G$, and assume that $|K|$ and $|G/K|$ are relatively ...
I'm trying to prove that if $|G|=30=2\cdot 3\cdot 5$ then $G$ has a normal $3$-Sylow or a normal $5$-Sylow. By the Sylow theorems, we would argue by contradiction in order to prove it cannot be $n_3=... 1answer 17 views ### Restricted linear representations of abelian groups If$G$is a group (say finite for simplicity though the question applies to infinite groups as well), what can one say about the subgroup$G^*_n = \text{Hom}(G, \mu_n)$of the group of all linear ... 1answer 33 views ### Which groups have only real representations? An irreducible representation$\rho$(with character$\chi$) of a finite group is called a "real" representation if its Frobenius-Schur indicator is 1:$$\frac{1}{\lvert G \rvert} \sum_{g \in G} \chi\... 1answer 53 views ### Conjugacy classes of$\mathcal D_{10}$. I was wondering if there is a special technique to find the conjugacy classes of$\mathcal D_{10}=\left<a,b\mid a^5=b^2=1,bab^{-1}=a^{-1}\right>$, and of$\mathcal D_{2n}=\left<a,b\mid a^n=b^...
Schur's Lemma: Let $(\Pi_i,V_i)$, $i=1, 2$ be two irreducible representations of a group $G$, and let $\phi : V_1 \to V_2$ be an intertwiner. Then either $\phi = 0$ or $\phi$ is a vector space ...