For questions about Sylow theorems in the context of group theory.
3
votes
1answer
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sylow basis of finite solvable groups
Let $G$ be a finite solvable non-$p$-group and $A$ be a maximal subgroup of $G$.
Therefore $A$ is of primary index $p^{n} $, that is $|G : A|=p^{n}$ where ...
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
1
vote
1answer
38 views
$P \cap N$ is a Sylow $p$-subgroup of $N$, where $N$ is normal in $G$ and $P$ is a Sylow $p$-subgroup of $G$?
In 'A Course in Group Theory' by Humphreys, Proposition 11.14 says that if $G$ is a finite group, $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$, then $P \cap N$ is a Sylow ...
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4answers
113 views
Can a group of order 3000 be a simple group?
Can a group of order 3000 be a simple group? How about the case of a group of order 1000?
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1answer
69 views
Prove the intersection of a Sylow $p$-subgroup and a subgroup is the unique Sylow $p$-subgroup
The statement we need to prove is:
Let $P$ be a normal Sylow $p$-subgroup of $G$ and let $H$ be any subgroup of $G$. Prove that $P\bigcap H$ is the unique Sylow $p$-subgroup.
Can you give some ...
-3
votes
2answers
63 views
Is isomorphic to a Sylow $p$-subgroup means is a Sylow $p$-subgroup [closed]
$P \in Syl_p(G)$ and $g \in G$. If $gPg^{-1} \cong P$, does this imply $gPg^{-1} \in Syl_p(G)$ ? And why? Thanks.
4
votes
1answer
41 views
Examples of groups with a certain number of Sylow 2-subgroups
Let $G$ be a group of order 50 and $m$ be the number of Sylow 2-subgroups of $G$. What are the possible values of $m$? For each value in your list, give an example of a group $G$ for which $m$ ...
0
votes
1answer
25 views
Need to show that order of orbits under group action is non-trivial and intersection of two p-groups is a proper subgroup
I'm working my way through the second and third sylow theorems in my book. Here's the relevant bit:
We have a group $G$ of order $p^\alpha m$ where $p$ does not divide $m$. We have that $Q$ is a ...
1
vote
1answer
27 views
Question about Sylow's Theorem/Conjugation of the set of conjugates of P
I'm trying to understand a proof of the 2nd and 3rd parts of Sylow's Theorem. In some preliminary work, my book establishes that $P$ is a Sylow p-subgroup of $G$.
Then it defines ...
6
votes
1answer
73 views
Groups with 20 Sylow subgroups
Is there a reasonably easy proof that a finite group with exactly 20 Sylow $p$-subgroups has PSL(2,19) or PGL(2,19) as a quotient group?
What if we weaken this to merely: “a group of order 760 has a ...
0
votes
0answers
31 views
Use of inclusion/exclusion when determining the groups of a given order
In my group theory class, when we are asked questions like
Determine the groups of order 30
or
Prove no group of order 90 is simple
We can do this, using a combination of Sylow's theorems ...
2
votes
1answer
56 views
Group of order 100: 1 or 3 subgroups of order 50?
Show that every group of order 100 has a subgroup of order 50. Show also that the
number of subgroups of order 50 is either 1 or 3.
For the first part I did the following:
As $|G|=100=2^2 5^2$ ...
3
votes
1answer
62 views
There exist Sylow subgroups $P$ and $Q$ for which $[P:P \cap Q]=[Q : P \cap Q] = p$.
From D&F's sylow theory section:
Show that if $n_p\not\equiv 1 \mod p^2$ then there are distinct Sylow $p$-subgroups $P$ and $Q$ of $G$ for which $[P:P \cap Q]=[Q : P \cap Q] = p$.
Are ...
3
votes
1answer
66 views
Cyclic Sylow $p$-subgroups are central in their normalizer when $p$ is the smallest prime divisor of $|G|$.
Let $p$ be the smallest prime dividing the order of a finite group $G$. If $P$ in $\operatorname{Syl}_p(G)$ and $P$ is cyclic, prove that $N_G(P)=C_G(P)$.
This is not homework. It is from ...
2
votes
1answer
82 views
About the proof that a simple group of order 60 is isomorphic to A5
I am stuck proving that a simple group $G$ of order $60$ is isomorphic to $A_5$.
In particular:
I have shown $|Syl_5(G)|=6$ and $|Syl_3(G)|=10$, so there must be $6\cdot(5-1)=24$ elements of order ...
3
votes
1answer
68 views
Orders of centralizers $C_G(g)$ in a group of order 60?
Given a group $G$ of order 60 with 24 elements of order 5, 20 of order 3, and 15 of order 2, how do we find the sizes of centralisers of elements of $G$ without proving $G\simeq A_5$?
By considering ...
2
votes
1answer
77 views
A group of order 30 has a normal 5-Sylow subgroup.
There are several things that confuse me about this proof, so I was wondering if anybody could clarify them for me.
Lemma Let G be a group of order 30. Then the 5-Sylow subgroup of G is
normal.
...
1
vote
2answers
88 views
Sylow Theorems Question
Let $p$ be the smallest prime that divides the order of a finite group $G$. If $H$ is a Sylow $p$-subgroup of $G$ and is cyclic. Prove that $N(H)=C(H)$.
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votes
0answers
89 views
Finding Sylow Subgroups.
Finding all the Sylow subgroups of a group is a problem known in computational group theory. If we were given a set of generators $g_1, \ldots g_k$ for each of $n$ subgroups of a group $G$ and we want ...
3
votes
1answer
64 views
Sylow $p$-Subgroups of Classical Groups over $\mathbb{F}_p$
Let $p$ be a prime, and let $G$ be any of the finite classical groups $SL_n(\mathbb{F}_p)$, $O_n(\mathbb{F}_p)$, or $SP_n(\mathbb{F}_p)$. Let $P$ be a Sylow $p$-subgroup of $G$. What is $P$ as a ...
4
votes
1answer
57 views
Sylow $p$-subgroups of $SO_3(\mathbb{F}_p)$
Let $G=SO_3(\mathbb{F}_p)$. We have $|G|=p(p^2-1)$. Let $n_p$ be the number of Sylow $p$-subgroups of $G$. Is it true that $n_p=p+1$?
We have the two conditions
$n_p\equiv 1\mod p$
$n_p\mid ...
1
vote
1answer
63 views
what does “the conjugacy part of sylow's theorems” denote?
$|G|=p^{a}m$
the conjugacy part of sylow's theorems , what does this denote? the first theorem ( the exist of sylow $p$-subgroup ) ? or the second one ( every $p$-subgroup is contained in some ...
2
votes
1answer
61 views
$p$-Sylow subgroups of $\operatorname{GL}(n, \mathbb{F}_{p^k})$
Is there a way to classify all $p$-Sylow subgroups of $GL(n,\, \mathbb{F}_{p^k})$ where $|\mathbb{F}_{p^k}|=p^k$? Clearly the prime $p$ (that is the characteristic of the base field) is a very ...
1
vote
3answers
115 views
On Groups of Order 315 with a unique sylow 3-subgroup .
in dummit and foote , an exercise asked me to prove that , if$ G$ is a group of order $315$ , $G$ has a normal sylow $3$-subgroup then , $G$ is abelian .
this is exercise number $27$ , section $5$ , ...
6
votes
2answers
125 views
If $P_1 , P_2 $ are two $p$-sylow subgroups, prove that $ P_1 \bigcap $ $P_2$ = $ { 1 } $
If $P_1 , P_2 $ are two sylow $p$-subgroups of the group $G$ prove that:
$ P_1 \bigcap $ $P_2$ = $ { 1 } $
I tried to prove it by induction as follows: proved it when $P_1 , P_2$ have the ...
4
votes
1answer
86 views
On Group of order $30$ and $60$.
In this question on yahoo answers ,
the answer says ,
"with $t = 6$, then there are 6 * (5 - 1) = 24 elements of order $5$ "
my question is , how did " 6 * ( 5 - 1 ) " come from ?
Which ...
1
vote
1answer
154 views
On the Group of order $pq$ where $p , q $ are primes .
Let $G$ be a group, $\lvert G \rvert = pq$ where $p$, $q$ are primes, $q$ is bigger than $p$.
Let $P$ be a Sylow $p$-subgroup and $Q$ be a Sylow $q$-subgroup
and let $n_p$= the number of Sylow ...
2
votes
1answer
80 views
Normal subsets of a Sylow p subgroup are conjugate if and only if they are $N_G(P)$ conjugate.
The following is a question from Dummit & Foote.
Prove that if $U$ and $W$ are normal subsets of a Sylow $p$-subgroup
$P$ of a finite group $G$ then $U$ is $G$-conjugate to $W$ if and only
...
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 ...
1
vote
1answer
162 views
Classify up to isomorphism the groups of order $203$
Classify up to isomorphism the groups of order $203$. Assume the face that the least $ k \geq 1$ such that
$$2^k \equiv 1 \mod{29}$$
is $k = 28$. [HINT: Look at the Sylow subgroups and ...
1
vote
2answers
119 views
Calculate number of Sylow 5 - subgroups in $S_{10}$. Prove every element of order 5 lies in some Sylow 5-subgroup
Calculate the number of $n_5$ of Sylow 5-subgroups in $S_{10}$ (a product of several integers is an acceptable answer) and check that $n_5 \equiv 1 mod 5$. Prove that every element of order 5, ...
0
votes
1answer
60 views
Describe Sylow 5 subgroup in $S_{10}$. Prove they're isomorphic to $C_5 \times C_5$. Prove every two subgroups are conjugate
Describe explicitly all Sylow 5= subgroups in $S_{10}$. Prove that every Sylow 5-subgroup is isomorphic to $C_5 \times C_5$. Prove that every two Sylow 5-subgroups are conjugate (explictiley, not ...
1
vote
1answer
168 views
Existence and structure of a group of order $p^2q$ where $p\mid q-1$ from a given presentation.
Let $p$ and $q$ be integer primes such that $p$ divides $q-1$.
(a) Show that there exists a group $G$ of order $p^{2}q$ with generators $x$ and $y$ such that $x^{p^{2}} =1$, $y^{q}=1$, and ...
4
votes
1answer
202 views
Constructing a Simple Group with a Specific Number of Sylow $p$-subgroups
I've been studying for my final exams, and I came across the following question:
If possible, give an example of a simple group $G$ with $n>1$ Sylow $p$-subgroups such that the order of $G$ ...
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 ...
