All Questions
53 questions
2
votes
0
answers
61
views
Suppose $P$ is a $p$-subgroup of the $p$-solvable group $X$. Prove $O_{p'}(N_X(P)) \subseteq O_{p'}(X)$.
I want to prove the following lemma
Suppose $P$ is a $p$-subgroup of the $p$-solvable group $X$. Then $O_{p'}(N_X(P)) \subseteq O_{p'}(X)$.
$O_{p'}(X)$ means the $p'$-core of $X$. That is, $O_{p'}(X)...
- 77
3
votes
2
answers
165
views
Sylow subgroups of $\mathrm{SL}_2(\mathbb{Z}_p)$ and $\mathrm{SL}_2(\mathbb{F}_p)$
Let $p,l$ be odd primes and $p\neq l$. I am interested in determining the $l$-Sylow subgroups of the groups $\mathrm{SL}_2(\mathbb{Z}_p)$ and $\mathrm{SL}_2(\mathbb{F}_p)$.
I have read that for $\...
- 1,270
1
vote
2
answers
105
views
Question regarding the properties of an automorphism group of a Sylow P subgroup
The context for this question has to do with proving: Groups of order $pq$ with $p < q$ have a normal subgroup of order $q$ and are cyclic iff $q$ is not congruent to $1$ mod $p$. I will leave out ...
- 87
3
votes
0
answers
88
views
Let $G$ be finite & nonsimple such that all normal subgroups of $G$ are $p$-groups for a given prime $p$. Is it true that $G$ is also a $p$-group?
Let $G$ be a finite non-simple group such that all normal subgroups of $G$ are $p$-groups for a given prime $p$. Suppose also that the $p$-Sylow subgroups of $G$ are not normal. Is it true that $G$ is ...
- 307
3
votes
1
answer
122
views
Show that $q\mid |N_G(H\cap K)|$ where $H,K\in\mbox{Syl}_p(G)$ with $|G| = p^nq$.
Let $G$ be a group of order $p^nq$ where $p,q$ are distinct prime numbers and $n$ is positive integer. Suppose $G$ has at least two distinct Sylow $p$-subgroup $H'$ and $K'$ such that $H'\cap K'\neq 1$...
- 5,157
0
votes
0
answers
104
views
Groups of order $16$ and their subgroups of index $2$
Question: For each of the groups $P$ of order 16 determine the classes of subgroups of index $2$ under the action of $\operatorname{Aut}(P)$.
Context: For almost $60$ years it has seemed to me that on ...
2
votes
1
answer
267
views
Understanding the definition of Sylow $p$-subgroups
Here is the definition of Sylow $p$-group (source: wikipedia)
For a prime number $p$, a Sylow $p$-subgroup of a group $G$ is a maximal $p$-subgroup of $G$, i.e. a subgroup of $G$ that is a $p$-group (...
- 716
1
vote
1
answer
156
views
For a group $G$ of order $p^n$, $G\cong H$ for some $H\le\Bbb Z_p\wr\dots\wr\Bbb Z_p$.
This is Exercise 5.3.2 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. The result is mentioned in the following ...
- 47.2k
2
votes
1
answer
429
views
Understanding the normalizer of a Sylow $p$-subgroup
The proofs I have seen of the second and third Sylow theorems use the following result (which my professor stated is the key as to why the Sylow theorems work):
Lemma. Let $P$ be a Sylow $p$-subgroup ...
- 2,927
1
vote
0
answers
57
views
Let $|G|=p^7$ and $K\le G$ with $|K|=p^3$. Then $\exists M\le G$ of order $p^4$ such that $K \trianglelefteq M$.
Let $G$ be a group with order $p^7$ and $K\leq G$ with $|K|=p^3$. Then there exists a subgroup $M$ of $G$ of order $p^4$ such that $K \trianglelefteq M$.
My immediate thought is to consider the ...
- 6,653
0
votes
2
answers
476
views
Show that a group of order $175$ is not simple.
$|G| = 175 = 5^2 \times 7$
After small calculation I found that only possible value of $n_5 = 1$ and $n_7 = 1$.
How to prove that $G$ is not simple group?
-1
votes
1
answer
191
views
Let $p$ be a prime number and let $G$ be a $p$-group. Then $G$ contains a normal subgroup of order $p^k$ for every nonnegative $k\le r$
Let $p$ be a prime number, and let $G$ be a $p$-group: $|G|=p^r$ . Then $G$ contains a normal subgroup of order $p^k$ for every nonnegative $k\le r$
But are there any normal subgroup of order $p^n$ ...
- 173
3
votes
1
answer
149
views
Prove that any group of order $945$ has at least one subgroup of order $9$
I'm trying to solve this group theory problem from my abstract algebra course. It goes like this:
Prove that any group of order 945 has at least one subgroup of order
9.
First, I noticed that $945=3^...
- 3,609
4
votes
1
answer
254
views
Prove that any group $G$ with $|G|=588$ is solvable
I'm stuck trying to solve this problem from my abstract algebra course:
Prove that every group of order $588$ is solvable (If you assume that all groups of certain order are solvable, you must prove ...
- 3,609
5
votes
0
answers
358
views
Bounds for probability that two elements commute in a group?
Let $G$ be a finite non-abelian group, and lets randomly choose two elements of $G.$ It seems quit well known that the probability that they commute is at most $\text{Pr}(G)\le\dfrac{5}{8}.$ Here is a ...
- 18.7k
8
votes
1
answer
347
views
A problem from Isaacs's Finite Group Theory
I was revisiting group theory in detail and reading Isaacs's Finite Group Theory book in my own time. Sorry that I am asking an exercise question but this is the one I am stuck completely. Any help ...
- 3,581
3
votes
1
answer
110
views
Group of order $q^3p^3$, where $p,q$ are twin primes greater than $10$, is solvable
Let $q>p>10$ be twin primes, i.e., $q=p+2$. Show that every group of order $q^3p^3$ is solvable.
This should be proven without using Burnside's theorem. Looking at the Sylow $p$-subgroup and ...
- 1,404
8
votes
1
answer
234
views
A simple group with $|\operatorname{Syl}_p G| \le 6$ is cyclic
Let $G$ be a simple, finite group, s.t. for every prime $p$, it satisfies $k_p=\left|\operatorname{Syl}_p G\right| \le 6$. Show that $G$ is cyclic.
My attempt: Let $n=p_1^{e_1}p_2^{e_2}\ldots p_r^{...
- 1,404
2
votes
2
answers
93
views
Proving $P$ is a Sylow $p$-group of $PN$
I am having trouble solving the following problem:
Let $G$ be a finite group of order $p^an$, where $p$ is a prime and $p \nmid n$.
Let $P$ be a Sylow $p$-group in $G$ and let $N \unlhd G$.
It can ...
- 185
2
votes
1
answer
112
views
Cardinality of direct product of Sylow $p$-subgroups
Let $G$ be a finite group such that, for all prime number $p$, $P_p$ is a normal Sylow $p$-subgroup of $G$. Let $I$ denote the set of prime numbers dividing $|G|$ and
$$K=\bigcup_{n\in\mathbb{N}}\{g\ |...
- 23
1
vote
0
answers
223
views
Quotient of quotient groups and Sylow $p$-subgroups
Let $G$ be a finite group and $N$ a normal subgroup of $G$. Let $\pi:G\rightarrow G/N$ be the canonical homomorphism.
Suppose $P$ is a Sylow $p$-subgroup of $G$. Then $\pi[P]\leq G/N$. Note that $G$ ...
- 11
1
vote
0
answers
97
views
Subgroup generated by Sylow $q$-groups of a finite group
Let $G$ be a finite group. Let $\mathfrak{P}$ denote the set of prime numbers and $n$ the order of $G$. Since $n>0$, there exists a unique family $(\nu_q(n))_{q\in\mathfrak{P}}$ of elements of $\...
- 11
3
votes
0
answers
211
views
Every finite group contains a Sylow $p$-subgroup
Let $G$ be a finite group of order $n$ and $p$ a prime number. Write $n=p^rm$ for some $r\in\mathbb{N}$ and $m\in\mathbb{N}_{\geq1}$ such that $m\not\in p\mathbb{Z}$. Let
$$E=\{X\subset G\ :\ |X|=p^r\}...
- 31
2
votes
1
answer
116
views
On generalized fitting subgroup
I can't understand so much the second paragraph (page 160) of the proof of the lemma 31.17(1) (pages 160) in M. Aschbacher, Finite Group Theory about generalized fitting subgroup. Here I post the ...
- 563
3
votes
3
answers
316
views
If $P\le G$ is a sylow-$p$ and $Q$ is any $p$ subgroup, then $Q\cap P = Q\cap N(P).$
If $P\le G$ is a sylow-$p$ and $Q$ is any $p$ subgroup, then $Q\cap P = Q\cap N(P).$
I'd appreciate any help.
I have a proof from some old notes but it says that it is sufficient to prove that if $...
- 597
1
vote
1
answer
74
views
Non-abelian finite group as a direct product of $p$-groups
My professor has given us some review material to help us prepare for our final. One of the questions is as follows:
Let $M$ be a finite abelian group of order $|M| = \prod\limits_{i = 1}^{l} p_i^{...
- 81
0
votes
1
answer
71
views
Group of order $2^2 \cdot 3^n$, $n \geq 1$.
Let $G$ be a group of order $2^2 \cdot 3^n$, with $n \geq 1$. The objective is to show that there is a normal subgroup of order $3^n$ or $3^{n-1}$; I can do the first part using the Sylow Theorems, ...
0
votes
0
answers
38
views
Finite Sylow subgroups of periodic groups
Let $G$ be a (possibly infinite) periodic group, and suppose that $G$ admits a maximal finite $p$-subgroup $P$. By this I mean that we do not assume that $P$ is not strictly contained in any other $p$-...
- 3,085
0
votes
0
answers
85
views
If $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$
In 'Finite Groups' by Gorenstein, it is stated that if $P$ is a Sylow $p$-subgroup of $G$ then there exists a normal subgroup $K$ such that $G/K$ is isomorphic to $P/P \cap G'$′. The proof is the ...
1
vote
2
answers
1k
views
Number of normal subgroups of order $p^s$ of a $p$-group
Let $G$ be a $p$-group. Show that the number of normal subgroups of $G$ that have order $p^s$ is $1$ $mod(p)$.
I think I have to use Sylow theorems and the fact that every subgroup of order $p^s$ is ...
- 963
1
vote
0
answers
26
views
Why is $M(p)/L_p $ a Dedekind p-group? (Where $M(p)/L$ is the p-Sylow of $G/L$)
Let $G$ be a finite $T$-group (i.e. a group in which every subnormal subgroup is normal). Assume that $L$ is a normal abelian subgroup of odd order of $G$ and suppose that $G/L$ is a Dedekind group. ...
- 127
1
vote
1
answer
205
views
Intersections of $p$-Sylow subgroups of group of order $p^2q$
Let $G$ be a group of order $p^2q$ ($p$ and $q$ are distinct primes) and $P_1$ and $P_2$ be distinct
$p$-Sylow subgroups of $G$. Then if $Z(G) = \{e\}$, $P_1\cap P_2 = \{e\}$
How do I go about ...
- 1,335
1
vote
0
answers
196
views
Normal p-subgroup and maximal subgroup
Let $G$ be a finite group and $p$ a prime s.t. $p\big||G|$, and let $P$ be a normal p-subgroup of $G$, with $|P|=p^m$. I want to prove the following:
If $M$ is a maximal subgroup of $G$, then $P\...
- 1,251
1
vote
3
answers
1k
views
About the definition of Sylow p-groups
Here is the definition of Sylow p-group (source: wikipedia)
For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e. a subgroup of G ...
user370967
3
votes
2
answers
2k
views
How do we know all Sylow $p$-subgroups of a given order are distinct?
Right now, I am in the process of showing that a group $G$ of order $35$ must be cyclic, for the purpose of then showing that there is a unique group of order $35$, namely $\mathbb{Z}_{35}$.
To that ...
user100463
2
votes
0
answers
154
views
Group of order $3^k5^l$ is solvable
So I'm not sure I'm proving this right and would appreciate a correction if needed.
Let $G$ be a group of order $3^k5^l$, where $k,l\in\mathbb N$ and $k\le 3$. Prove that $G$ is solvable.
Proof:
...
- 189
2
votes
2
answers
202
views
Number of $p$-groups of a given size in a Sylow subgroup
Problem: Let $p$ be a fixed but arbitrary prime number and $a$ a non-negative integer. Let $P$ be a Sylow $p$-subgroup of $G$.
Then prove that the number of subgroups of $G$ of order $p^a$ which are ...
- 3,753
1
vote
1
answer
290
views
$|G|=pq$, where $p$ and $q$ are primes, is the semidirect product of subgroups of orders $p$ and $q$
I am currently working on the following exercise:
Let $G$ be a group of order $pq$, where $p$ and $q$ are primes and $p > q$. Prove that $G = N \rtimes H$ for some subgroups $N$ and $H$ of ...
user100463
3
votes
2
answers
5k
views
A finite $p$-group cannot be simple unless it has order $p$
I am to trying to prove this theorem: A finite $p$-group cannot be simple unless it has order $p$.
I have this:
Let $G = P$ and $|G|=p$; then there exists $N$, a normal subgroup of $G$ by Lagrange'...
- 93
1
vote
1
answer
67
views
When does prime $p$ divide a term in the numerator and denominator of ${p^m \cdot k}\choose {p^m}$ the same number of times?
This has questions comes to me via a proof to Sylow's First Theorem where $G$ is a finite group of order $p^m \cdot k$, the number $p$ is a prime divisor of $|G|$, and $p^m$ is the highest power of $p$...
- 706
0
votes
1
answer
1k
views
The order of the normalizer of a $p$-subgroup of $S_{p}$ [closed]
I found it In Exercise in abstract algebra by Dummit and Foote.
Let $P$ be a Sylow $p$-group of $S_p$.
What is the order of $N_{S_p}(P)$?
- 1,441
7
votes
2
answers
1k
views
Classify all groups of order $p^2q^2$ up to isomorphism
Let $p,q \in \mathbb{N}$ be prime numbers with the properties
$2 < p < q$ and $q - 1 , q + 1 \notin \left\langle p \right\rangle$
Classify all groups of the order $p^2q^2$ up to isomorphism.
...
- 532
2
votes
1
answer
282
views
p-element centralizing a Sylow p-subgroup
Let $G$ be a finite group, $P$ a Sylow $p$-subgroup for a prime $p$ and
$g$ a $p$-element with $gxg^{-1} = x$ for all $x \in P$. Then $g \in Z(P)$.
Is this true? How can i prove that $g \in P$ ?
- 245
9
votes
1
answer
8k
views
Difference between definitions of $p$-subgroup and Sylow $p$-subgroup
I'm reading Abstract Algebra by Dummit and Foote and the following
definitions are made:
$1$. A group of order $p^{\alpha}$ for some $\alpha\geq1$ is called a $p$-group. Subgroups of $G$ which are $p$...
- 23.3k
0
votes
1
answer
29
views
$|\{ x\in X: g.x=x \space\space\space \forall g\in G \}| = |X|\space mod \space p$
Let $G$ be a p-group. $|G|=p^n$ for some n.
Let X be a finite set so that $\,p\nmid |X|\,$,
G acts upon X.
Denote $A:= \{ x\in X: g.x=x \space\space\space \forall g\in G \}$
I am trying to show $|...
- 165
14
votes
1
answer
6k
views
Group of order $p^{n}$ has normal subgroups of order $p^{k}$
Q: Prove that a subgroup of order $p^n$ has a normal subgroup of order $p^{k}$ for all $0\leq k \leq n$.
Attempt at a proof: We proceed by Induction. This is obviously true for $n=1, 2$.
Suppose it ...
- 1,686
0
votes
1
answer
393
views
Sylow subgroups of soluble groups
Suppose $G \leqslant S_p$ acts transitively on $\{1,...,p\}$ for prime $p$. Let $P \leqslant G$ be a Sylow p-subgroup. Is it true that $G$ is soluble <=> $P \triangleleft G$?
- 251
0
votes
1
answer
697
views
Quotient groups of $p$-groups
Suppose I am trying to show that a group $G$ is solvable and I gotten to having $Z(G)$ be a p-group and $G/Z(G)$. Now if I can show that $G/Z(G)$ is also a $p$-group, then both are solvable implying $...
- 241
4
votes
1
answer
456
views
Existence of a normal subgroup in a finite group.
Let $G$ be a finite group. If a Sylow $p$-subgroup $P$ of $G$ is contained in the centre, then does there exist a normal subgroup $N$ of $G$, such that $P \cap N = \{e\}$ and $PN=G$?
Thanks in ...
- 2,221
3
votes
2
answers
548
views
$H$ must contain every Sylow $p$-subgroup of $G$
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 contain every Sylow $p$-subgroup of $G$.
- 31