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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24 views

Let G be a group of order p^n suppose that H is a normal subgroup of G [closed]

Prove that there exists a normal subgroup H' a subset of G such that [H':H] = p
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28 views

Product $PN$ of normal subgroups is abelian

I am trying to show that every non-abelian group $G$ of order $6$ has a non-normal subgroup of order $2$ using Sylow theory. First, Sylow's Theorem says the number of Sylow $2$-subgroups $n_2$ is ...
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Upper-triangular matrices as union of centralizers of elements

Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z} )$ such that $n\leq p$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. The ...
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1answer
51 views

Let $|G|=2^np$, $p$ an odd prime, $H\unlhd G$ a Sylow $2$-subgroup with $H\cong(\Bbb{Z}/2\Bbb{Z})^n$, $p\nmid 2^n-1$. Prove $Z(G)$ is nontrivial.

Let $G$ be a group with $|G|=2^np$ ($p$ an odd prime). Let $H$ be a normal Sylow $2$-subgroup such that $H\cong(\mathbb{Z}/2\mathbb{Z})^n$. Prove that if $p$ does not divide $2^n-1$, then $G$ has a ...
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Order of a subset

Hihi, im in a trouble with a excercise. I hope someone can help me: Let (G, $\bullet$) be a finite group, and U a subset of G. $\\$ Let $\phi$ : G $\times$ U $\to$ U be a group action such that $\...
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1answer
48 views

Proving that a simple group of order $360$ has 10 Sylow $3$-subgroups and that their pairwise intersection is trivial

I have that $G$ is a simple group of order $360$. By the 2nd, 3rd Sylow theorems we know that there are $n_3 = 1+3k, k \in \mathbb{Z}$ Sylow 3-subgroups and that $n_3$ divides $2^3 \cdot 5$, since $...
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1answer
59 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 ...
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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-...
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1answer
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About finite group such that $Z(G)$ must be cyclic

Let $G$ be finite group and $G$ has at least one sylow subgroup $K$ such that $K\ntriangleleft G$ . proof if every nonnormal and abelian subgroup of G be cyclic then $Z(G)$ is cyclic. Theorem 1:...
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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^{...
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1answer
36 views

Order of finite quotients of a group

Let $G$ be an infinite residually finite group, and let $p$ be a prime. What are some examples where the finite quotients of $G$ have no elements of order $p$? Equivalently, what are some examples ...
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1answer
45 views

$|G|=105$, Show that is $P_3$ is a Sylow-3-subgroup then $5||N_G(P_3)|$

$|G|=105$, Show that is $P_3$ is a Sylow-3-subgroup then $5||N_G(P_3)|$ This question is given as an exercise here. I am having a hard time seeing it. From my Sylow's Theorem training I know that the ...
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1answer
41 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, ...
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1answer
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$G$ a group of order 210. If $G$ is commutative show it is cyclic. If $G$ is any group show it contains a subgroup of index 2.

$G$ a group of order 210. Part 1 If $G$ is commutative show it is cyclic. Consider the prime factorization $210=(2)(3)(5)(7)$. By Sylow's theory we know that Sylow subgroups exist of order 2, 3, 5, ...
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101 views

Galois Group of $x^{6}-2x^{3}-1$

I was trying to compute the normal closure of $\mathbb{Q}[\alpha]$, where $\alpha = \sqrt[3]{1+\sqrt{2}}$. I had a reallyyyy hard time proving that $x^{6}-2x^{3}-1$ is irreducible. I proved that it ...
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Find the Sylow $2$-subgroup of $G$. [closed]

Set $G = (Z/6Z) \times (Z/18Z)$. Since $G$ is abelian, then there is only one Sylow $p$-subgroup for each positive prime $p$ that divides $|G|$. Find the Sylow $2$-subgroup of $G$. Having some ...
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1answer
26 views

group with $1+rp$ Sylow $p$-subgroups and existence of group in $\mathrm{Sym}(1+rp)$ with $1+rp$ Sylow $p$-subgroups

Let $r\in \mathbb{N}$ and $p$ a prime. Suppose that a group $G$ has $1+rp$ Sylow $p$-subgroups. Then there exists $H\leq \mathrm{Sym}(1+rp)$ that has precisely $1+rp$ Sylow $p$-subgroups. I was ...
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1answer
37 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 ...
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1answer
59 views

Show there are are no simple groups of order 1638

I'm trying to use Sylow Theory to show there are no simple groups of order 1638. I got so far as to factor $1638 = 2*3^2*7*13$ and compute that we must have $n_2|819$ $n_3 \in \{1,7,3\}$ $n_7 \in \{1,...
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1answer
47 views

Every subgroup of order 60 contains a subgroup of index 5

Aluffi IV.2.25 (in the chapter on Sylow theorems) suggests the following exercise: Assume $G$ is a simple group of order $60$. Use Sylow's theorems and simple numerology to prove that $G$ ...
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the subgroups of special linear group

Let $G={\rm PSL}(2,8)$. So $|G|=2^3 \times 3^2 \times 7$. Let $ P \in {\rm Syl}_2(G)$, $Q \in {\rm Syl}_3(G)$ and $R \in {\rm Syl}_7(G)$. I know $P \cong C_2 \times C_2 \times C_2$ and $Q,R$ are ...
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45 views

Prove that $G = \{e_G, a, a^2 , a^3 , . . . , a^{p−1} , b, ab, a^2 b, . . . , a^{p−1 }b\}.$

Let $G$ be a group of order $2p$ where $p$ is a positive odd prime. By the $1^\text{st}$ Sylow Theorem, there exist $a, b \in G$ such that $|a| = p$ and $|b| = 2.$ Prove that $G = \{e_G, a, a^2 , a^3 ,...
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1answer
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Existence of $A \in \operatorname{Syl}_p(H)$ and $B \in \operatorname{Syl}_p(K)$ such that $AB \in \operatorname{Syl}_p(G)$

$\DeclareMathOperator{\Syl}{Syl}$Assume that $G = HK$ is a finite group, where $H$ and $K$ are two subgroups of $G$. I want to find an $A \in \Syl_p(H)$ and $B \in \Syl_p(K)$ such that $AB \in \Syl_p(...
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1answer
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Groups of order $pq$, $p$ and $q$ primes with $p<q$.

RIGHT ANSWER BY DUMMIT: Here Suppose $|G|=pq$ for primes $p$ and $q$ with $p<q$. Let $P \in Syl_{p}(G)$ and let $Q \in Syl_{q}(G)$. We are going to show that $Q$ is normal in $G$ and if $P$ also ...
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1answer
29 views

Invariant Sylow subgroups

Today, I'm reading lemma 2.2.c of an article by Antonio Beltran. Lemma 2.2. Suppose that A is a finite group acting coprimely on a finite group G, and let $C = C_G(A)$. Then, for every prime p, ...
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1answer
30 views

Fixed points of the conjugation action of a p-Sylow subgroup on the set of p-Sylow subgroups

Aluffi IV.2.10 suggests the following exercise. Let $P$ be a $p$-Sylow subgroup of a finite group $G$, and act with $P$ by conjugation on the set of $p$-Sylow subgroups of $G$. Show that $P$ is the ...
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2answers
50 views

Prove that $K$ is a Sylow $p$-subgroup of $T$.

Let $G$ be a finite group and let $T$ be a subgroup of $G.$ Suppose $K$ is a Sylow $p$-subgroup of $G$ such that $K \subseteq T.$ Since $K$ is a Sylow $p$-subgroup of $G$, then $|K|=p^m.$ I want to ...
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1answer
30 views

Prove that the number of subgroups of $G$ with order $p^r$ is equivalent to $1 \pmod{p}$ by using group action.

Claim. Let $G$ be a group, $p$ be a prime number and $r \in \Bbb{N}$ such that $p^r$ divides $|G|$. If $n_G(p^r)$ is the number of subgroups of $G$ with order $p^r$. Show that $n_G(p^r) \equiv 1 \pmod{...
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1answer
34 views

Subnormal $\pi$-groups of a finite group $G$ are contained in $O_{\pi}(G)$

Let $G$ be a finite group. I want to prove that If $N$ is a subnormal $\pi$-subgroup of $G$, then $N\le O_{\pi}(G)$. I first tried the case where $\pi=\{p\}$, $p$ a prime, but I got stuck. Here ...
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2answers
87 views

Number of elements of order $2$ in a group of order $10$.

Consider a group $G$ of order $10$. Then $G$ can be abelian as well non-abelian. What is the number of non-trivial elements of $G$ of order $2$? Answer: If $G$ is abelian, $G$ can be cyclic as ...
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1answer
26 views

Clarification on the proof that the homomorphic image of a Sylow p-subgroup is Sylow p-subgroup.

I realize this question has been asked 3 times now on the site, but I cannot understand the first line of the most popular proof: Homomorphic image of a Sylow p-subgroup is Sylow p-subgroup. The ...
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1answer
55 views

Why exactly elements in different Sylow subgroups can't commute in $A_5$?

My professor remarked that: There is no room in $A_5$ for anything to commute. Or certainly not for elements in different Sylows to commute. Can someone please explain why "there is no room" in $...
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62 views

Decomposing the alternating group $A_5$ as “product” of its Sylow subgroups

My professor remarked in class today that the alternating group $A_5$ can be decomposed as a product of its Sylow subgroups but only if we align the Sylows in a specific way and combine them in the ...
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1answer
37 views

Understanding a Lang's proof of Sylow theorem (related to the class formula)

This proof comes from S. Lang's Algebra Revised Third Edition. Honestly 2/3 of this proof makes sense to me, except the application of class formula and the followings. My questions are: Why do we ...
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Prove that there is no simple group of order $216=2^{3}\cdot 3^{3}$. [duplicate]

Prove that there is no simple group of order $216=2^{3}\cdot 3^{3}$. I feel like I am supposed to use the Index Theorem here but when I use Sylow's Third Theorem I have that $n_{3}\in\{1,4\}$. I am ...
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2answers
51 views

Finite p-group contains normal subgroups $H_i$, $|H_i| = p^i$

Let $G$ be a finite group of order $p^n$, where $p$ is prime. Show that $G$ contains normal subgroups $H_i$ for $1 \leq H_i \leq n$ such that $|H_i| = p^i$ and $H_i < H_{i+1}$ for $1 \leq i <...
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Is there an elegant method to show the alternating group $A_5$ as disjoint union of its Sylow subgroups and the trivial element?

Per the discussion in the comments of this question the alternating group $A_5$ is apparently the disjoint union of its Sylow subgroups and the trivial element. I know it can be done explicitly by ...
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0answers
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How to show that $A_5$ has no subgroup of order 15 and 20 but has a subgroup of order 12 *using* Sylow' theorems?

So we just started learning about Sylow theorems in class. This PDF has a proof of the fact that "the alternating group $A_5$ has no subgroup of order $15$ and $20$ but it does have a subgroup of ...
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1answer
67 views

What should be the mental process for quickly evaluating representative Sylow subgroups of $A_5$?

This Groupprops Wiki page has a nice chart classifying the subgroups of $A_5$ upto automorphism. It shows the various representative subgroups. However, say if I were told to manually find the ...
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1answer
34 views

Showing the injectivity of the following homomorphism (direct product of normal Sylow subgroups)

Consider normal Sylow subgroups $G_1, \ldots, G_r$ of a finite abelian group $G$. Let's set up the homomorphism $\phi: G_1 \times \ldots \times G_r \to G$ s.t. $(g_1, \ldots, g_r) \mapsto g_1\ldots ...
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2answers
123 views

Frattini sugroup and normal subgroup

For any group $G$, let $\Phi(G)$ denote the Frattini subgroup of $G$. Let $G$ be a finite group, such that $\dfrac{G}{\Phi(G) } \cong K \times \mathbb{Z}_{p}$, where $p$ is prime number. Prove that ...
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2answers
31 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 ...
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1answer
48 views

Show that a group of order 12 cannot have nine elements of order 2.

This is what I have so far. Assume the contrary, then by Sylow's Third Theorem, the number of Sylow 3-subgroups is either 1 or 3. If it were 1 then there are only 7 elements of order 2. Thus there are ...
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1answer
41 views

What are the possibilities for the number of elements of order $5$ in a group of order $100$?

I want to make sure that I am doing this right. Since $100 = 5\cdot 20$, by Sylow's Third Theorem we have that $n_{5}\equiv 1$ (mod 5) and $n_{5}\mid 20$. So $n_{5}\in\{1,6,11,16,21,26,\dots\}$ and ...
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1answer
51 views

Question about Sylow-subgroups

Here : https://groupprops.subwiki.org/wiki/A5_is_the_unique_simple_non-abelian_group_of_smallest_order it is shown that the only simple subgroup of order $\ 60\ $ upto isomorphy is $A_5$ I do not ...
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1answer
31 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\ |...
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1answer
32 views

Let $G$ and $H$ be finite groups and let $p$ be a positive prime number that divides the order of $G$ and that divides the order of $H.$

Let $S$ be a Sylow $p$-subgroup of $G$ and let $T$ be a Sylow $p$-subgroup of $H$. Prove that $S×T$ is a Sylow $p$-subgroup of $G× H$. Suppose $S \times T$ is a Sylow $p$-subgroup. Then $S \times T = ...
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1answer
27 views

Intersection of Sylow subgroups is reduced to $\{e\}$

Suppose $H$ is a finite group. Let $p,q$ be two primes and $S_q,S_p$ the associated Sylow $p$-subgroups of $H$. If $p\ne q$, then $S_q\cap S_p=\{e\}$. Suppose $p\ne q$. Let $|S_p|=p^r$ and $|S_q|=...
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0answers
48 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 $G$ ...

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