Articles of group theory

Non-abelian group of order $p^3$ without semidirect products

I am trying to read a proof that there are at most two non-abelian groups of order $p^3$ if $p$ is an odd prime. The proof presents it as two cases: the first, where every non-identity element has order $p$, is fairly straight forward and results in the group generated by $a, b, c$, each […]

Maximal normal $\pi$-subgroups and torsion subgroups

Let be $\mathbb{R}$ the real numbers and $\mathbb{Z}$ the integers. Let $G = \mathbb{R}/\mathbb{Z}$. Determine $O_\pi(G)$ – the normal maximal $\pi$-group of $G$, $T(G)$ – torsion subgroup.

Question about homomorphism of cyclic group

if $\varphi: G\to H$ is homomorphism. How do I prove that if $a\in G$ have finite order so $\varphi(a)$ had finite order to, and that:$$ord(\varphi(a))\mid ord(a)$$ Thank you!

Computing square roots implies factoring $n = pq$

I’m proving that computing square roots in $\mathbb{Z}_{pq}$ implies factoring $n = pq$ with $p,q$ primes. The solution give you an algorithm: repeat 1. pick y from {1,…,n-1} 2. x = y^2 mod n 3. y’ = random square root of x until y’ != y and y’ != – y mod n Basically the […]

Classifying $\mathbb{Z}_{12} \times \mathbb{Z}_3 \times \mathbb{Z}_6/\langle(8,2,4)\rangle$

I wish to classify $\mathbb{Z}_{12} \times \mathbb{Z}_3 \times \mathbb{Z}_6/\langle(8,2,4)\rangle$ according to the fundamental theorem of finitely generated abelian groups. We have that it is of order $72$. Based on previous help received here, I have attempted to set it up as a matrix: $$\begin{bmatrix}12 & 3 & 6\\8&2&4\end{bmatrix}$$ But I am unable to find any […]

Unique intermediate subgroup and double coset relation I

Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is it true that $HgK=KgH$, $\forall g \in G$ ? Experiment (GAP) : It’s true if $[G:H] \le 30$ and $\vert G/H_G \vert \le 10000$. ($H_G$ is […]

Quotient of nilpotent group is nilpotent

Edit: I managed to rephrase my proof in a way that does not resort to coset multiplication. I think the resulting proof is better. I’ve added it as an answer below, while preserving the original question to avoid wrecking the context. In his book Finite Group Theory, section 1.D, Isaacs mentions without proof the following […]

How to show that the orbits of the action of Gs on S \ {s} have lengths that are equal in pairs.

Question:Let G be a group of odd order acting transitively on a set S. Fix s ∈ S. Show that the orbits of the action of Gs on S \ {s} have lengths that are equal in pairs. My idea: set a point a$\in$S \ {s}, then the order of the stabilizer of a in […]

What can $ab=b^2a$ and $|a|=3$ imply about the order of $b$ when $b\neq e$?

Possible Duplicate: Let$G$ is a group, $a$ and $b$ are non-unit elements of $G$, $ab=bba$. … Let $G$ be a group and $a,b\in G$ such that $$ |a|=3, ab=b^2a, b\neq e. $$ What can I say about $|b|$? What I get so far is something like $$ ba^2=a^2b, ab^2=b^4a. $$ I suspect that one can […]

The intersection of two Sylow p-subgroups has the same order

Let $G$ be a finite group and assume it has more than one Sylow $p$-subgroup. It is known that order of intersection of two Sylow p-subgroups may change depending on the pairs of Sylow p-subgroups. I wonder whether there is a condition which guarantees that intersection of any two Sylow $p$-subgroups has the same order. […]