4
votes
2answers
94 views
Finite abelian $2$-group
If $G$ is a finite abelian group such that $o(x)=2$ for all $x \neq e$ and $|G|=2^n$ for some $n\in\mathbb N$, prove that $G \cong \mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$ ($n$ factors).
Any ...
2
votes
2answers
48 views
Rotman's Introduction to to the theory of groups. Exercise 3.45.
Can you give me a hint on the first part of the exercise?
Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
1
vote
2answers
75 views
$H$ must contain every Sylow $p$-subgroup of $G$
G is finite and has a prime factory. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that
0
votes
0answers
47 views
Generators in $p$-groups
Let $G$ be a non-abelian finite $p$-group such that $G/[G,G]$ is elementary abelian group of order at least $p^3$. Let $S=\{\bar{x_1},\bar{x_2},\bar{x}_3,\cdots,\bar{x}_n\}$ be independent generators ...
2
votes
2answers
70 views
$G$ $p$-group. If $H\triangleleft G$, then $H\cap C(G)\ne \{e\}$
I'm pretty new on this subject and I need a hint to begin to solve this question:
If $G$ is a finite p-group, $H\triangleleft G$ and $H\ne \{e\}$, then
$H\cap C(G)\ne \{e\}$
Thanks for any ...
1
vote
3answers
44 views
If H is a p-group, the order of any H-orbit is a power of p.
This comes from the proof of the third Sylow Theorem in Michael Artin's "Algebra".
Let S be the set of Sylow p-groups in a given group G of order $p^em$. Let H be any Sylow group. If we decompose S ...
1
vote
4answers
59 views
How do I show that $N\leqslant Z(G)$ without using Sylow theorems?
Let $G$ be a group of order $p^2$ ($p$ being a prime) and $N$ be a normal subgroup of $G$ such that $O(N)=p.$ Without using the fact any group of order $p^2$ is abelian or sylow theorem how to show ...
2
votes
1answer
48 views
What are central automorphisms used for?
A central automorphism is an automorphism $\theta$ for which $x^{-1}\theta(x)\in Z(G)$ for each $x\in G$.
It's not difficult to prove that the set of central automorphisms forms a subgroup of ...
2
votes
5answers
79 views
Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)
I want to show without using Sylow theorem that
Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)
My attempt: Since $|G|=2p,$ even $\exists~a\neq e\in G$ such that ...
4
votes
2answers
69 views
$p$-Group as union of subgroups
It is well known that a group can not be union of two proper subgroups. For finite $p$-groups, we can say more:
A finite $p$-group can not be union of $p$ proper subgroups.
Moreover,
...
4
votes
2answers
83 views
If every element has prime power order and $Z(G) \neq 1$ then $G$ is a $p$-group.
In a finite group $G$ if every element is of some prime power order (prime may vary with element) and if $G$ has non trivial center then prove that $G$ is actually of prime power order.
Deduce that ...
2
votes
1answer
46 views
About finite $p$-group finitely generated.
Let $p$ a prime number. Let $G$ be a $p$-group finitely generated, say, $G = \left<x_1, x_2, \ldots, x_m \right>$. Denote by $\gamma_i(G)$ the $ith$ term of the lower central series of $G$. ...
1
vote
2answers
69 views
irreducible representation of non-abelian p-group
Can someone help with the following problem?
Let $G$ be a non-abelian group of prime-power order $p^n$ and $E$ be an irreducible $G$-space over $\mathbb{C}$ giving a faithful representation of $G$.
...
3
votes
1answer
36 views
Maximal Subgroups Containing given Element
Let $G$ be an elementary abelian $p$-group of finite rank, and $1\neq g\in G$. How do we parametrize the maximal subgroups of $G$, which contain $g$?
1
vote
3answers
39 views
A question on finite $p$-groups. [duplicate]
Is true that if $G$ is a $p$-group finite, say, $\mid G \mid = p^d$, then $G$ is $d$-generated?
1
vote
2answers
107 views
Let $G$ be a finite group and $P$ be a Sylow $p$-subgroups of $G$. Let $H$ be a subgroup of $G$ such that $N_G(P)\subset H$. Prove that $N_G(H)=H$
1.Let $G$ be a finite group and $P$ be a Sylow $p$-subgroups of $G$. Let $H$ be a subgroup of $G$ such that $N_G(P)\subset H$. Prove that $N_G(H)=H$
2.Let $G$ be a group of order $143$. Show that ...
4
votes
3answers
53 views
at least one element fixed by all the group
$G$ is a p-group and $S$ is a set that $G$ acts on. p does not divide $|S|$. Why is there at least one element $a\in S$ such that $|O(a)|=1$, or in other words, $G_a=G$?
I tried to ask this question ...
4
votes
2answers
55 views
Is it true to say $Z(G)\subseteq N_G(H)$?
Let $p$ be an prime number and $G$ a group of order $p^n$ and $H$ be subgroup of $G$ of order $p^{n-2}$ and which is not normal in $G$. Is it true to say that $Z(G)\subseteq N_G(H)$?
13
votes
1answer
109 views
The equivalence classes of $N\sim M\Leftrightarrow G/N\cong G/M$.
Let $G$ be a finite group. Given some $N\unlhd G$, define $$\mathfrak{C}_N:=\{M\unlhd G : G/M \cong G/N\}.$$
How are the subgroups in $\mathfrak{C}_N$ related? Is there some other description ...
3
votes
2answers
68 views
$\pi$-radical of group
Let $G$ abelian and periodic. Let $\mathbb{P}$ be the set of prime numbers, $\pi
\subseteq \Bbb P$ and $\pi ^{\prime }=\Bbb P\setminus\pi $.
Let $O_{\pi }\left( G\right) =\left\langle ...
11
votes
0answers
80 views
References on the theory of $2$-groups.
Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to ...
6
votes
4answers
155 views
There exists only two groups of order $p^2$ up to isomoprhism.
I just proved that any finite group of order $p^2$ for $p$ a prime is abelian. The author now asks to show that there are only two such groups up to isomorphism. The first group I can think of is ...
3
votes
2answers
100 views
Subgroups of order $p$ and $p^{n-1}$ in a group of order $p^n$.
I have a group $G$ of order $p^n$ for $n \ge 1$ and $p$ a prime. I am looking for two specific subgroups within $G$: one of order $p$ and one of order $p^{n-1}$. I don't think I would use the Sylow ...
4
votes
3answers
134 views
Abelian $p$-group with unique subgroup of index $p$
Let $G$ be a finite abelian $p$-group with a unique subgroup $H$ of index $p$. It is a fact that $G$ is cyclic. This can be deduced from the classification theorem for finite abelian groups by writing ...
7
votes
0answers
96 views
Generic properties of $p$-groups
I have the impression that most people in algebra believe the following statement to be true, but I have no reference for it.
Fix a natural number $n$. Consider for each prime $p$ the set of all ...
6
votes
2answers
321 views
Classify groups of order 27
Let $|G|=27$. Prove that all subgroups of index 3 are normal. Classify all groups of order 27.
I can do the first one, but the classification is overwhelming. I don't even know where to start.
...
2
votes
2answers
154 views
$p$-group and normalizer
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 $G$.
...
3
votes
2answers
143 views
A normal subgroup $H$ with $[G:H]$ coprime to $p$ contains every Sylow $p$-subgroup of $G$.
I was working on the following question:
Let $p$ be a prime factor of the order of a finite group $G$. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that H must ...
4
votes
2answers
88 views
On inner automorphisms group of a $p$ group
Let $G$ be a $p$-group of class 2 and $\exp(\operatorname{Inn}(G))=p^c$. Then prove $\frac{G} {Z (G)}$ has the form $C_{ p^c}\times C_{ p^c}\times C$ for some (possibly trivial) abelian $p$-group ...
45
votes
0answers
857 views
+50
Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G \,\,\,s.t.\,\,\,H\cong G/N$?
A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization for groups for which this ...
7
votes
2answers
120 views
Very generic question about Commutator and Center
Let $G$ be a finite group and $G'$ the commutator group of $G$.
What can I say about $G' \cap Z(G)$?
Could you be as specific as possible about p-Groups?
5
votes
2answers
141 views
Nonabelian $p$-groups all of whose proper subgroups are abelian.
Theorem.
Let $G$ be a finite, non-abelian $p$-group all of whose proper subgroups are abelian. Then $|G'|=p$.
Take a counterexample of minimal order. Assume that exist a $H$ such that ...
7
votes
1answer
95 views
Bound on the number of p-groups for fixed exponent
It's well-known that for each prime number $p$ there are exactly two groups of order $p^2$, five of order $p^3$, and fifteen of order $p^4$ (at least when $p>3$).
I know that the classification of ...
1
vote
1answer
131 views
A question about intersection of center and commutator subgroup
Let $G$ be a finite group such that $G'\cap Z(G)\neq 1$. Suppose also that $G'$ is an elementary abelian $p$-group; $G'\nleq Z(G) $; $(G/Z(G))'$ is a minimal normal subgroup of $G/Z(G)$.
Can we ...
48
votes
2answers
4k views
More than 99% of groups of order less than 2000 are of order 1024?
In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024.
Is this for real? How can one deduce this result? ...
2
votes
2answers
147 views
Does a Frobenius group with a $p$-group complement necessarily have a normal Sylow $2$-subgroup?
Let $G=KH$ be a Frobenius group of even order with Frobenius kernel $K$ and Frobenius complement $H$ such that $\pi(H)=\{p\}$, where $p$ is prime. Why is a Sylow $2$-subgroup of $G$ normal in $G$?
2
votes
1answer
87 views
Tate $p$-nilpotent theorem
Tate $p$-nilpotent Theorem. If $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$ such that $P \cap N \leq \Phi (P)$, then $N$ is $p$-nilpotent.
My question is the following:
If ...
0
votes
1answer
89 views
If $a^{3^3}=b^9=1$ for generators $a,b$ of $G$, can we conclude that $G$ is a $3$-group?
I know that $a$ and $b$ are generators of a group $G$ and $a^{3^3}=b^9=1$.
Are these informations sufficient to affirm that the group is a $3$-group?
Adding the relation $b^{-1}ab=a^4$, can we ...
5
votes
1answer
159 views
Normal Subgroups in a p-group
How can one prove the following claim:
Elementary abelian $p$- group of order $p^n$ have the maximal number of normal subgroups among all $p$-groups of the same order.
Is is indeed true?
...
1
vote
1answer
102 views
Question About the Derived Series and Commutators
Given a p-group $F$ , we define the derived series as follows:
$ F^{(1)} = F$ , $ F^{(n)} = [F^{(n-1)} , F^{(n-1)} ] $ .
I'm now given the lower central series $F_n = [F,F_{n-1} ] $ ( $ F_1 =F , $ ) ...
5
votes
1answer
96 views
Is this problem correct that $HG'=G$?
Here, I have the following homework:
Let $G$ is a finite $p-$group and let $H$ be a subgroup of it such that $HG'=G$. Prove that $H=G$ ($G'$ is the commutator subgroup).
I have tried to show ...
