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Questions tagged [sylow-theory]

For questions about Sylow theorems in the context of group theory. Not for use with questions regarding Sylow systems, which belong in solvable-groups.

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1answer
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Finding number of subgroups of order 3 in an abelian group of order 24

Problem Find the number of subgroup of order 3 in an abelian group of order 24? Attempt Let $n_3$ be the number of subgroup of order 3 . Then by Sylow's theorem $n_3|8$ and $n_3 \equiv 1mod 3$. From ...
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1answer
38 views

Group with order $30$ is not a simple group.

Could someone prove the sentence given above in the title? I know that Sylow theorems should be used here. Let $N_{p}$ stands for number of $p$-Sylow subgroups in group $G$. I tried to use ...
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37 views

A question about Sylow p-subgroups in finite groups. [on hold]

Let $G$ be a finite group and $K$ a normal $p'$-subgroup of $G$. Let $\alpha : G \rightarrow G/K$ be the canonical surjection. Let $T$ be a Sylow $p$-subgroup of $G/K$ and $P$ a Sylow $p$-subgroup of $...
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An argument from Herstein [on hold]

The following is an argument from the sylow section of Herstien(topics in algebra). I’m not able to follow why $r$ is the highest power of p which divides the given binomial coefficient. Can someone ...
3
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1answer
33 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 ...
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1answer
65 views

Group of order 60

Let $G$ be a simple group of order $60$: i) Find the number of Sylow $3$- and $5$-subgroups of $G$. ii) Show that the alternating group $A_5$ has a subgroup of order $12$. iii)Show that $G$ is ...
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1answer
68 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 $G$ ...
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2answers
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Sylow Subgroups and Conjugation

Suppose that $G$ is a finite group whose order is divisible by a prime $p$. Let $S$ be the set of Sylow $p$-subgroups of $G$; let $H$ be an element of $S$. $H$ acts on $S$ by conjugation. The fact ...
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Find the number of $2$-Sylow subgroups in $GL_2(\mathbb F_5)$

First, we can calculate the order of $GL_2(\mathbb F_5)$ to find the order of the $2$-Sylow subgroups. $$|GL_2(\mathbb F_5)|=(5^2-1)(5^2-5)=24\cdot 20=2^5\cdot 3\cdot 5$$ Thus we have $|P|=2^5$. ...
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1answer
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The normalizer of a $p$ sylow subgroup is itself

By the orbit-stabilizer theorem since the action is transitive then an orbit $\{gPg^{-1}: g\in G\} =n_p$ is equal to the number of $p$ sylow subgroups in a group $|G|=p^{\alpha}s$ with $(p^\alpha,s)=1$...
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2answers
87 views

Show that a group of order $90$ is solvable

Let $G$ be a group of order $90$. Show that $G$ is solvable. Given that $90=2 \cdot 5 \cdot 3^2$ is of the type $pqr^2$, with $p, q,$ and $r$ primes, this case is not straightforward. My attempt to ...
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1answer
18 views

In a group of order 400, is the normalizer of one of the 16 Sylow 5-subgroups itself?

In a group $G$ of order $400 = 2^4 \cdot 5^2$, assume there are sixteen Sylow 5-subgroups. Let $P_5$ be one of them. Is the normalizer $N_G(P_5)=P_5$? I think this is true, as the order of $P_5$ is ...
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2answers
72 views

Show that a group of order 66 has a normal subgroup of order 33.

This question is somewhat similar to: A group of order $66$ has an element of order $33$. However, I do not understand how I would show that the subgroup of order 33 is normal. So far I have that ...
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41 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 ...
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1answer
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Abelian normal subgroups of A-groups

Let $G$ be a finite solvable group, where every Sylow subgroup is abelian. I want to show that if $A\lhd G$ is an abelian normal subgroup, then $$ A=(A\cap Z(G))(A\cap G')$$ This is easy if $A$ is a ...
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0answers
35 views

Show that any finite nilpotent group of square free order is cyclic.

Show that any finite nilpotent group of square free order is cyclic. Hint: Suppose G is such a group. Any Sylow subgroup of G is of prime order. Hint: Any finite nilpotent group is the direct ...
2
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1answer
74 views

How can $SN_G(D)=G$ if $S$ is not normal in $G$?

Is there any theorem which says anything like $SN_G(D)=G$ where $S$ is a $p$-Sylow subgroup and $D$ is the intersection it has with some other subgroup? I know the first that will probably spring to ...
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87 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. ...
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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 ...
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1answer
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In a group of order $m p^n$ for $p$ prime, if $k<n$, is there an element of order $p^k$? [duplicate]

Let $G$ a group of order $mp^n$ where $p$ is prime. Let $k\leq n$. Is there an element of order $p^k$ ? Since $p$ divide $|G|$, by Cauchy theorem, there is $g\in G$ s.t. $g$ has order $p$. I can't do ...
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Number of proper nontrivial subgroups for a group $G$ with size $pq$?

If I have a group $G$ with order $pq$ with $p,q$ primes, $p\neq q$. Then I want to try to apply Sylow's third theorem. I want to argue that there is only 2 proper subgroups of $G$. Since $p$ ...
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1answer
28 views

G has 4-Sylow 3-subgroups

I am working on the following problem: Prove that if $\lvert G \rvert = 12$ and $G$ has $4$ Sylow 3-subgroups, then $G \equiv A_4$. If you let $G$ acts by conjugation on the set containing the $4$ ...
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2answers
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Group Theory Sylow Subgroup [closed]

What's an example of a group $G$ and an integer $n$ dividing $|G|$ with $0 < n < |G|$ such that $G$ has no subgroup of order $n$.
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1answer
57 views

Every Sylow subgroup is normal, then $G$ has a subgroup of order $m$ for every division $m$ of $|G|$

Some trouble working out an algebra problem. Suppose that every Sylow subgroup of a finite group $G$ is normal. Prove that $G$ has a subgroup of order $m$ for every divisor $m$ of $|G|$.
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45 views

No group of order 10000 is simple

A proof of this fact was already given here: No group of order 10,000 is simple However, I am wondering whether or not the following proof works as well: By way of contradiction, suppose $G$ is ...
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27 views

The cases in proving that a group of order 90 is not simple

I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that lets us count elements and get a ...
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1answer
43 views

Group Theory Subgroups Sylow Theory [closed]

Let $G$ be a finite group, $p$ a prime and $e$ a nonnegative integer. If $p^e$ divides the order $|G|$ of $G$, show that $G$ has a subgroup of order $p^e$.
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Group of order greater than 8 doesn't decompose into a direct product and Sylow 2-subgroup isomorphic quaternion group

Is there a group of order greater than 8 that does not decompose into a direct product such that its Sylow 2-subgroup isomorphic quaternion group $Q_8$?
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Question about p-Sylow subgroups being maximal

I am having some trouble wrapping my head around p-Sylow subgroups at the moment. I am given that for p, prime, a p-Sylow subgroup of G is a maximal p-subgroup that is not a proper subgroup of any ...
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1answer
62 views

Group of order $6p^m$ is solvable for prime $p\geq 7$

Let $p\geq 7$ be a prime and $m$ be a positive integer. Prove that group of order $6p^m$ is solvable. Attempt: By Sylow's theorems we have that $n_p \mid 6$ so $n_p\in \{1,2,3,6\}$ where $n_p$ is ...
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1answer
61 views

Proof that no group of order $525$ is simple

I would like some verification that any group $G$ of order $|G| = 525 = 5^2 \cdot 3 \cdot 7$ is not simple. I've attached my argument below. Please let me know if you see any issues. Thanks Let $G$ ...
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1answer
27 views

Let H be a Sylow p-subgroup of G. Prove that H is the only Sylow p-subgroup of G contained in N(H).

Let H be a Sylow p-subgroup of G. Prove that H is the only Sylow p-subgroup of G contained in N(H). I saw a proof online that was pretty long, but can't I just argue that if $H \subset N(H)$, then $H$...
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2answers
39 views

Group Isomorphism regarding Sylow Subgroups

Suppose I have given two groups say, $G_1,G_2$ such that they have same order.I'm assuming they are non commutative.Then their Syllow subgroups has same order clearly.If I'm given that the number of ...
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$gcd(p,m-1)=gcd(p-1,m)=1$ if and only if G has a normal Sylow p-subgroup in $Z(G)$

Suppose that $|G|=pm$ where $p$ is a prime and $p\not|m$. Prove that $gcd(p-1,m)=gcd(p,m-1)=1$ iff G has a normal Sylow p-subgroup in $Z(G)$ I have a problem in both directions. How can I solve ...
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2answers
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Is $n_p(G)$ unique for different groups of size $p$?

If $G$ has that $|G| = p^am$ where $p$ is prime and $gcd(p,m) = 1$, we have that $n_p(G)$ counts of Sylow $p$-subgroups in $G$. If $P$ is a Sylow $p$-subgroup, by the third Sylow theorem \begin{align*...
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Group Theory (Sylow p-subgroups questions)

Let $GL_n\mathbb{F}_p$ be the group of invertible $n \times n$ matrices with entries coming from $\mathbb{F_p}=\{0, 1, ..., p-1\}$ and with group operation multiplication of matrices. (We are writing $...
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1answer
28 views

Sylow's First Theorem acting on Abelian Group

Background In the book of Judson's book on abstract algebra, Sylow's First Theorem is proved by first invoking the class equation and then considering the case where $p$ can/cannot divide $[G:C_G(g)]$...
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1answer
54 views

Equivalent conditions for a Group $G$ with order $p^2q$ ( with $p>q$ both prime) be abelian.

I saw this homework many times, but always asks in the statament that $p^2 \not\equiv 1$ (mod $q$) and $q \not\equiv 1$ (mod $p$). But today in a book text of Galois theory I Saw a similar example ...
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1answer
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Relationship between prime power and the divisor

Background When trying to prove that a group G has a non-trivial centre if its order is a power of a prime $p$, there involves a step in which we claim the number of left cosets of a centraliser ...
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1answer
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About Sylow subgroup and internal direct product.

Let $G$ be a finite group and $P$ a Sylow $p$-subgroup of $G$. Let $Q$ be a subgroup of $P$. If $Z(Q)$ is a Sylow $p$-subgroup of $C_G(Q)$, is $C_G(Q)$ a internal direct product of $Z(Q)$ and $O_{p'}(...
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Let $p$ be a prime dividing $|G|$, and let $S_p$ be a $p$-Sylow subgroup of $G$. Show that $N(N(S_p))=N(S_p)$.

Attempt: Trivially $N(S_p) \subseteq N(N(S_p))$ because a subgroup is a subset of its normalizer. Then, let $g\in N(N(S_p))$. Then we have $gN(S_p)g^{-1}=N(S_p)$ by the definition of normalizer. But $...
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228 views

How big must the union of a group's Sylow p-subgroups be?

For various orders $n$ it's a common exercise to prove that a finite group $G$ of order $n$ can't be simple by using the Sylow theorems to show that there is some prime $p \mid n$ such that the number ...
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70 views

proving that a group of order 60 is simple using homomorphisms.

I want to show that the group G, where $|G|=60$ and has 20 elements of order 3 is simple. Here's what I did: Suppose that G is not simple this implies that $n_3>1, n_2>1, n_5>1$ where these ...
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Is a group with order $3^3\cdot 5\cdot 7$ possible?

Is the group $|G|=3^3\cdot 5\cdot 7$ possible? I've been examining it, and it seems like there can't be enough elements in the Sylow-p-subgroups. I.e. there can only be $21$ (or $1$) groups of ...
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1answer
50 views

Does the centraliser of the intersection of any two Sylow $p$-subgroups contain all Sylow $p$-subgroups?

Consider a collection of Sylow $p$-subgroups. If any two of these intersect non-trivially then they are both contained within the centraliser of their intersection. Now assume that $P_1$ and $P_2$ ...
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67 views

Help me to better understand the intersection of Sylow $p$-subgroups?

Say you have some collection of Sylow $p$-subgroups. For example lets say their are 7 Sylow $3$-subgroups. and that each has order $9$ There are three cases( right ?) i) any pair $P_1\neq P_2 $ ...
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1answer
47 views

Are any two distinct p-Sylow subgroups normal?

This is very clear that if we have unique $p$-Sylow subgroup in a group G then it is normal in G by using second Sylow theorem, as single $p$-Sylow subgroup in a group is self conjugate to itself.... ...
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1answer
32 views

Groups of order $p^xq^y$ where $x,y>1$ are integers and $p,q$ are distinct primes

How does one prove that a group of order $p^xq^y$ where $p,q$ are distinct primes and $x,y>1$ are integers, is NOT simple, without using Burnside's theorem or solvability? I think one way to ...
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2answers
58 views

Cyclicity of group of order $455$

The following excerpt is from my lecture notes and I have understood almost everything except one moment. Prove that any group of order $5\times 7\times 13$ is cyclic. Using Sylow's theorem we ...
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1answer
75 views

Number of $p$-subgroups of finite group

Assume we have a finite group $G$, $|G|=p^cm$ where $p\not\mid m$ is prime. Fix a subgroup $H$ of order $p^a$ and a number $a\le b\le c$. Prove that the number of subgroups of order $p^b$ which ...