What does “finite rank” mean in the context of divisible abelian $q$-group? A divisible abelian $q$-group of finite rank is always a Prüfer $q$-group or it can be also a finite product of Prüfer $q$-group? Thanks for all future answers to my questions.

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 […]

Here is the question: a) Show that if $p$ is a prime number and $P$ is a $p$-subgroup of a finite group $G$, then $[G:P]=[N_G(P):P]$(mod p), where $N_G(P)$ denotes the normalizer of $P$ in $G$. b) Assume that $H$ is a subgroup of a finite group $G$, and that $P$ is a Sylow $p$-group of […]

I am trying to solve an exercise in D. Robinson’s book A Course in the Theory of Groups, which asks me to show that if $G$ is polycyclic and residually finite p-group for infinitely many prime p, then $G$ is nilpotent and finitely generated torsion-free. How do I show the nilpotent part? The hint given […]

How can I create a random group of order $4096$ with a center of size $2$ ? The algorithm should be able to create every possible group with the given properties in principle. I think the list of groups with the required properties is far too large and probably not even known. I tried semidirect […]

A p-group is a group of order $p^d$ where p is a prime. If the center has order $p^m$ (since its order must divide the order of the group) then we have a one dimensional faithful irreducible representation of the center which would map a generator of the center to $e^{2\pi i/m}$. Could we then […]

Let $G$ be a finite 2-group such that $\left|\dfrac{G}{Z(G)}\right|=4$, $Z(G)$ is not cyclic and $Z(G)$ has at least one element of order 4. Then prove that there exists an automorphism $\alpha$ of $G$ such that $\alpha(z)\neq z$ for some $z\in Z(G)$. Do the proof in My attempt 2 is true? My attempt1: Let $\dfrac{G}{Z(G)}=\{Z(G), aZ(G), […]

I’m looking for as much information about the orders of elements in minimal generating sets of finite abelian $p$-groups as possible. What I really need is complete knowledge about the possible orders of elements in such groups and how many of each order there can be. I know that every group of the form $(\mathbb […]

From a little reading, I know that for $p$ and odd prime, there are two nonabelian groups of order $p^3$, namely the semidirect product of $\mathbb{Z}/(p)\times\mathbb{Z}/(p)$ and $\mathbb{Z}/(p)$, and the semidirect product of $\mathbb{Z}/(p^2)$ and $\mathbb{Z}/(p)$. Is there some obvious reason that these groups are nonabelian?

Prove that if $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m <n.$ Since $G$ is of prime-power order I know $|Z(G)| \ne e$ so there is an $a\in Z(G)$ with order $p$ such that $p \mid |Z(G)|$. Now, the subgroup generated by is normal since it’s a subgroup […]

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Line integration in complex analysis
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Example of set which contains itself