All Questions
30 questions
0
votes
1
answer
78
views
Some questions about the proof of the third Sylow theorem [closed]
Sylow's Theorem: Let $|G| = p^n · m$ with $p$ prime, m coprime to $p$, and $n ≥ 1$. The number $α(p)$ of $p$-Sylow groups of $G$ is a divisor of $m$ and of the form $α(p) = 1+kp$ for a $k ≥ 0$.
Proof:...
- 892
1
vote
1
answer
134
views
Grillet - Abstract Algebra - Second Edition, proposition 5.5 of section II.
I'm having trouble understanding the following proof of this proposition:
If a Sylow $p$-subgroup of a finite group $G$ is normal in $G$, then it is the largest $p$-subgroup of $G$ and the only Sylow ...
- 2,235
1
vote
0
answers
107
views
Theorem 10, Section 2.5 of Hungerford’s Abstract Algebra
Theorem 5.10. (Third Sylow Theorem) If $G$ is a finite group and $p$ a prime, then the number of Sylow $p$-subgroups of $G$ divides $|G|$ and is of the form $kp+1$ for some $k\geq O$.
Proof: By the ...
- 4,508
0
votes
1
answer
113
views
Theorem 7, Section 2.5 of Hungerford’s Abstract Algebra
Theorem 5.7. (First Sylow Theorem) Let $G$ be a group of order $p^nm$, with $n\geq 1$, $p$ prime, and $(p,m)=1$. Then $G$ contains a subgroup of order $p^i$ for each $1\leq i\leq n$ and every subgroup ...
- 4,508
1
vote
1
answer
177
views
Why can't a group of order 132 contain 3 Sylow $2$-subgroups and be simple?
In the question titled Prove that if |G|=132 then G cannot be simple, it is shown a group of order $132$ cannot be simple. My summary of the proof:
assume group is simple
deduce the number of Sylow $...
- 1,031
3
votes
1
answer
145
views
Proof of Sylow's first theorem conclusion.
$\newcommand{\fix}{\text{Fix}}$
$\newcommand{\orb}{\text{Orb}}$
$\newcommand{\stab}{\text{Stab}}$
I'm having an issue concluding the proof of Sylow's 1st theorem in the case where we assume $p \nmid |...
- 2,724
1
vote
1
answer
179
views
Proof of 1st Sylow Theorem by induction on $|G|$.
Theorem: Let $G$ be a finite group and $p$ be a prime dividing the order of $G$. Then $G$ contains a Sylow $p$-subgroup.
I'm getting stumped at one part of the proof.
Proof: Let $G$ be a finite ...
- 2,724
3
votes
1
answer
46
views
Why is $|H \cdot gP|=1$?
In the begining of the Second Sylow Theorem
proof my teacher did the following:
Let $G$ be finite a group, $P$ a Sylow $p$-subgroup of $G$ and $H$ any $p$-subgroup. Consider the following group ...
- 5,109
3
votes
2
answers
904
views
If $G$ is a group of order $1575$ with normal Sylow $3$ subgroup, then show that Sylow $5$ and $7$ subgroups are both normal.
This is Dummit & Foote's Exercise 4.5.28 of "Abstract Algebra".
If $G$ is a group of order $1575$ with normal Sylow $3$ subgroup, then show that Sylow $5$ and $7$ subgroups are both ...
- 5,157
0
votes
3
answers
159
views
Understanding a proof that $|G|=pq,(p<q) \Rightarrow G$ is not simple.
Proposition
Let $G$ be a finite group and $|G|=pq$ where $p<q$ and $p,q$ are prime. Then, $G$ is not simple.
The questioneer of this page A finite group of order $pq$ cannot be simple. writes the ...
- 3,354
3
votes
2
answers
671
views
Prove that if $|G| = 160$, $G$ is not simple.
I'm trying to prove this with Sylow's Theorem. I understand that the intersection between two Sylow-2 subgroups $H$ and $K$ cannot be of order $16$, since $| H \cap K| = 16$ implies $H \cap K \lhd G$. ...
- 431
2
votes
2
answers
342
views
Proof of Sylow's Theorem (Herstein) - why is $no(H) = o(G)$?
The theorem is:
(Sylow's theorem): If $p$ is a prime number, and $p^\alpha |o(G)$, then $G$ has a subgroup of order $p^\alpha$.
Right before the proof, the author has established that if $n = p^\...
- 14.5k
0
votes
1
answer
274
views
Prove that any $p$-subgroup of a finite group $G$ is contained in a Sylow $p$-subgroup
I'm trying to understand the proof from this post:
Show a certain group is contained in a Sylow p-group.
But certain things I cant decipher. Specifically, I'm looking at the top-rated answer.
If these ...
- 597
0
votes
0
answers
50
views
Let $G$ be a non-nilpotent group where all the non-normal abelian subgroups of $G$ are cyclic. Then $G$ has cyclic center.
Theorem : Let $G$ be a non-nilpotent group such that all the non-normal abelian subgroups of $G$ are cyclic. Then $G$ has cyclic center.
Proof. Suppose that $Z(G)$ is non-cyclic. since $G$ is non-...
- 4,747
3
votes
1
answer
292
views
Clarification on proof of fundamental theorem of finite abelian groups
Herstein's Topics in Algebra provides a proof of the fundamental theorem of finite abelian groups, that is, every finite abelian group is the direct product of cyclic groups.
In an earlier exercise, ...
- 623
1
vote
1
answer
106
views
Clarification on partitioning a group into cosets
I'm reading I. N. Herstein's proof of Sylow's third theorem:
Theorem: The number of $p$-Sylow subgroups in $G$, for a given prime, is of the form $1+kp$.
Here is a picture of the proof, for ...
- 623
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
1
vote
0
answers
87
views
Prove that a group of order 45 is necessarily abelian by Sylows' Theorems [duplicate]
So far, my attempt of proof looks like this:
$\#G =45=3^2\times 5$, so:
By the 1st Sylow Theorem that G has p-subgroups of order 5, 9 and 3.
By the 3rd theorem, the p-subgroups of order 5 and 9 have ...
- 442
4
votes
2
answers
425
views
Group of order $p^{\alpha}q$ is not simple.
$|G|=p^{\alpha}q$, where $p,q$ are distinct primes, $\alpha \geq 1$. Show $G$ is not simple.
I am trying to follow a proof and I understand all of it except one part which is blocking me.
The proof ...
- 981
1
vote
1
answer
133
views
Why does $a^p \equiv 1\ (\text {mod}\ q)$?
Suppose $G$ is a group of order $pq$ with $p<q, p \nmid q-1$ and $p,q$ are primes. Then $G$ is cyclic.
The way our instructor has proved this theorem is as follows $:$
He first proved that $G$ ...
- 4,892
0
votes
0
answers
127
views
trying to understand this proof of sylows theorem that say the number of p-sylow subgroups is 1+kp
I'm Very confused as to what my lecturer means in the final few lines of a proof of one of sylows theorems means. The theorem in question is the one that says the number of sylow P-subgroups is 1+kp.
...
- 2,835
1
vote
0
answers
75
views
a few questions about what's going on in this proof of sylow's theorem I found
Note: If someone wants to even just answer my first question in the comments until someone else decides to give a full answer I'd be pretty happy. I just want to know there's no mistakes in it before ...
- 2,835
1
vote
1
answer
99
views
First Sylow Theorem proof [duplicate]
Let $G $ be group such that $p^a $divides $|G|$ then G has subgroup of order $p^a|$.
Proof:Let $|G|=p^am$
Let $\mu$ is set of all subset with $p^a$ elements. SO there are $\binom{p^am}{p^a}$ element ....
- 6,773
1
vote
1
answer
732
views
Understanding Proof of Sylow theorem from Herstein
I was trying to learn group theory indepedently form Herstein' Topics of Algebra.
I that I now reading Sylow's theorem.In that I trying to understand first proof of sylow theorem. But I am not able to ...
- 6,773
3
votes
0
answers
321
views
Explanation of a proof of the Second Sylow Theorem (Conjugation of Sylow p-subgroups)
I am an undergrad Mathematics student and I've been reading some additional literature for my lectures and came upon a quite short and seemingly elegant proof of the Second Sylow Theorem. Though, I ...
- 85
2
votes
1
answer
183
views
Finitely many conjugacy classes of finite subgroups of given order
Let $G$ be a periodic locally soluble group with finite Sylow p-subgroups for all primes p.
It is know that in these conditions $G$ is residually finite. Moreover it can be proved that $G$ has only ...
- 4,220
2
votes
1
answer
207
views
Understanding Sylow's First Theorem Using Double Cosets
Remark on Double Cosets
Sylow's First Theorem
The above hyperlinks are on the proof I'm referring to.
The above is the proof of: All Sylow $p$-groups are conjugate.
It is assumed: Let $G$ be a ...
user333821
0
votes
1
answer
477
views
Proof explanation on a group of order $595$ having a normal Sylow $17$-subgroup. [duplicate]
Prove that a group of order 595 has a normal Sylow 17-subgroup.
The proof is as follows:
By Sylow, $n_{17} = 1$ or $35$. Assume $n_{17} = 35$. Then the union of the Sylow
$17$-subgroups has $561$...
- 5,072
1
vote
2
answers
125
views
Cyclic groups of order $pq$ proof question
Let $H$ be a Sylow $p$-subgroup of $G$ and let $K$ be a Sylow
$q$-subgroup of $G$. Sylow’s Third Theorem states that the number of
Sylow $p$-subgroups of $G$ is of the form $1+kp$ and divides $pq$....
- 5,072
1
vote
1
answer
489
views
Class equation and orbit stabilizer theorem
I was reading the proof of the following theorem but I cannot understand how to use the class equation as he wants me to.
Theorem
Suppose that $G=HK$ where $H$ is a normal locally finite $p'$-...
- 4,220