# 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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### Groups and Cosets

I am studying the proof of the third Sylow's Theorem and I dont get this: Let $G$ be a finite group, $H$ a $p$-Sylow subgroup of $G$, $N(H)= \{g \in G : gH=Hg \}$. Note that $G = \cup_{g \in G}AgA$ (...
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### Sylow 2-subgroup of Suzuki Group $Sz(8)$

I need to find the isomorphism class containing the Sylow 2-subgroup of the Suzuki group $Sz(8)$. Can anyone give a reference?
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### Is every infinite group nilpotent iff it is direct product of its sylow p-subgroups?

We know that every finite group is nilpotent iff it is direct product of its sylow p-subgroups.is this also true for infinite groups?
1 vote
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### $S$-semipermutable subgroups

A subgroup $H$ of a finite group $G$ is said to be $S$-semipermutable in $G$ if it permutes with every Sylow $q$-subgroup of $G$ for all primes $q$ not dividing $|H|$. Assume that $G$ is solvable and ...
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### Normalizers of Sylow subgroups.

If $G\neq \left \{ 1 \right \}$ is a finite solvable group, then there is at most one prime $p$ such that if $P\in \operatorname{Syl}_p(G)$, then $N_G\left ( P \right )=P.$ I think it's necessary to ...
1 vote
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I am trying to solve this question: Prove that a group of order $11 \times 2^{10}$ has a nontrivial proper normal subgroup. My trial By Sylow theorems I know that $n_2 \in \{1,11\}$ and $n_{11} \in \{... 2 votes 1 answer 52 views ### Show that$SL_2(F_3)/Z(SL_2(F_3)) \cong A_4$Show that$SL_2(F_3)/Z(SL_2(F_3)) \cong A_4$I know that$|SL_2(F_3)/Z(SL_2(F_3))|= 12$. Then if the quotient group has a normal subgroup of order$4$then it is isomorphic to$A_4$. Suppose that it ... 2 votes 0 answers 73 views ### Prove that groups of order 200 are solvable Suppose$G$is a group with$|G| = 200$. Since$200 = 2^3 \cdot 5^2$I've used Sylow's theorem to make two claims concerning$n_G(8)$, the number of Sylow 2-subgroups of$G$: $$n_G(8) \equiv 1 \mod ... 2 votes 1 answer 44 views ### Abelian group of order pq is a subgroup of S_{p+q} Let G be a finite abelian group of order pq, where p<q are both primes. I want to show that G is isomorphic to a subgroup of S_{p+q} (but is not isomorphic to any subgroup of order S_{p+... 5 votes 1 answer 98 views ### Quotient of the Galois group of a splitting field generated by two roots Let f \in \mathbb{Q}[x] be irreducible of degree p, where p is a prime. Let K be the splitting field of f and suppose that there are roots \alpha and \beta of f such that K = \mathbb{... 2 votes 1 answer 67 views ### Action of the normalizer of the complement Let G be a finite group, and let p be a prime. Let U be a Sylow p-subgroup of G. By the Schur-Zassenhaus theorem, there is a complement T to U in N_G(U). Observe that N_G(T) acts by ... 1 vote 1 answer 114 views ### Nonabelian group of order 28. Is there a nonabelian group of order 28 whose 2-Sylow subgroup is isomorphic to \mathbb{Z}/4\mathbb{Z}? My reasoning is that by Sylow's Theorem, there is a 2-Sylow subgroup of order 4. Since it is ... 0 votes 1 answer 62 views ### Intersection of all Sylow p-subgroups I wanted to prove that the intersection of all Sylow p-subgroups of a finite group G is a normal subgroup of G. Can someone enlighten me how is this implication possible: If an automorphism \... 4 votes 1 answer 67 views ### Number of Sylow p-subgroup of \mbox{SL}_n(\Bbb F_p). What is the number of Sylow p-subgroup of \mbox{SL}_n(\Bbb F_p) where \Bbb F_p is finite field of order p? This problem is known for \mbox{GL}_n(\Bbb F_p). By checking the order, strictly ... 2 votes 2 answers 64 views ### If G has no subgroup of index 2 and G\leq S_n, then G \leq A_n. I am currently reading Abstract Algebra by Dummit & Foote. Discussing some techniques about the Sylow theorems they prove the proposition mentioned in the title (Proposition 12 (1), p. 204, 3rd ... 0 votes 0 answers 27 views ### 2-Sylowgroup of A_6 I want to find a 2-Sylowgroup, i.e. a subgroup of order 8, of A_6. I think I can solve this problem by brute force but in order to avoid endless calculation I was wondering whether someone could ... 7 votes 0 answers 94 views ### Proof there are no perfect groups of order 3024 How can I prove that there are no perfect groups of order 3024? My attempt is the following: Each non-trivial finite perfect group admits a non-abelian simple quotient. This holds because if the ... 1 vote 0 answers 36 views ### No group of order 160 is simple [duplicate] I'm trying to prove that no group of order 160 is simple. The following is my approach. Let G be a group of order 160. Note 160 = 2^55. I can easily get that n_2 = 1 or 5 and n_5 = 1 or ... 1 vote 0 answers 45 views ### Let p be prime. Let S be a Sylow p-subgroup of G. If x\in G then xSx^{-1} is a Sylow p-group. [duplicate] Let p be prime. Let S be a Sylow p-subgroup of G. If x\in G then xSx^{-1} is a Sylow p-subgroup. My attempt: Let |G|=p^n\cdot m where p is prime, n,m\in\mathbb{Z} such that p\nmid ... 1 vote 2 answers 418 views ### Why are Sylow Theorems and Sylow subgroups significant? If one read's Gallian's Abstract Algebra book then they would find that the chapter for Sylow Theorem's is quite hyped up. However, I am unable to understand the big picture of why Sylow subgroups and ... 1 vote 1 answer 84 views ### Do isomorphic groups have the same number of Sylow p-subgroups? I think that isomorphic groups should have the same number of Sylow p-groups, but I am not sure why, I am a little stuck on this, I really don't know where to even begin, or if this is even true (... 15 votes 2 answers 281 views ### Can we construct a group with exactly k Sylow-Subgroups? Inspired by the answers given by these three questions (here, here, and here), what is the general solution for constructing a group with a specific number of Sylow subgroups? That is, given a prime ... 4 votes 0 answers 84 views ### Alternate proof of Sylow's First theorem. I am trying to prove first Sylow Theorem using the Lemma: if G is a finite group such that has a Sylow p-subgroup and H\subset G, then H has a Sylow p-subgroup. The way I want to go about ... 0 votes 0 answers 51 views ### Finite groups G in which maximal subgroups of Sylow subgroups are normal in G If G is nilpotent, then maximal subgroups of Sylow subgroups of G are all normal in G. So the class of groups those finite groups with property that maximal subgroups of Sylow subgroups are ... -1 votes 1 answer 45 views ### G is supersolvable if the maximal subgroups of Sylow subgroups of G are normal in G G is supersolvable if the maximal subgroups of its Sylow subgroups are normal in G. Then G' is nilpotent. If P is a non-normal Sylow subgroup of G, then why is P not contained in G'? 2 votes 1 answer 44 views ### Strategy for classifying some groups of order pqr - recognizing direct factors I've been reviewing some of my notes from an abstract algebra class that I took and have been thinking about/redoing some of the examples we did classifying groups of smallish order pqr. In ... 3 votes 1 answer 57 views ### non-abelian groups of order of product of 3 distinct primes Suppose that p,q,r are distinct primes such that q = 1 \pmod p pq = 1 \pmod r Consider the following group G of order pqr: 1 element must be the identity element Exactly q-1 elements are ... 0 votes 1 answer 149 views ### What is the question "If G is group of order 360 what are the possible isomorphism types of the Sylow p-subgroups for p=2,3,5,7?" really asking? What is the question "If G is group of order 360 what are the possible isomorphism types of the Sylow p-subgroups for p=2,3,5,7?" really asking? I am familiar with the Sylow theorems, but ... 1 vote 1 answer 54 views ### Number of subgroups of given order is a product of prime powers congruent to 1\pmod p If we have a group G, and maximal a prime power p^k divides |G| (meaning that p^{k+1} does not divide |G|), then we must have a subgroup H of order p^k, by Sylow's first theorem. Let ... 0 votes 1 answer 63 views ### Let G be a group of order 2014. Prove that G has a normal subgroup of order 19 and G is solvable [duplicate] Let G be a group of order 2014. Prove that G has a normal subgroup of order 19 and G is solvable. The first part directly follows from the Sylow Theorems, if you write 2014 = 2 \cdot 19 \... 1 vote 1 answer 66 views ### When is a Sylow p-subgroup normal? Let A_5 be the alternating group of degree 5. I would like to prove that the number s_5 of Sylow 5-subgroups of A_5 ist 6. With |A_5| = \frac{5!}{2} = 60 = 5 \cdot 12 and the Sylow theorems I ... 3 votes 0 answers 59 views ### Exponent of p-sylow subgroup of symplectic groups over the field GF(p) Let P be the p-sylow subgroup of symplectic group Sp(2n,p), where p is an odd prime. I want to know if there exists anything about the exponent of P? I know that Sp(m, p) \leq GL(m, p) ... 1 vote 0 answers 34 views ### Relation between p-Sylow subgroups of a group G and the p-Sylow subgroups of G/H [duplicate] I'm learning about the Sylow theorem and the p-Sylow subgroups. In certain applications of the Sylow Theorem one considers the p-Sylow subgroups of the quotient group G/H to conclude a statement ... 0 votes 1 answer 36 views ### Why are these two Sylow subgroups disjoint? I have the following problem: Let G be a finite group with card(G)=p^2q with p<q two prime numbers. We denote s_q the number of q-Sylow subgroups of G and similarly for p. I have ... 1 vote 0 answers 38 views ### Contradiction in the application of the Sylow Theorem on the alternating group A_5 Somehow I get a contradiction in the following application of the Sylow Theorem. If G is a group with cardinality 60 = 2^2 \cdot 3 \cdot 5 there are exactly 4 or 24 elements of order 5. I ... 2 votes 1 answer 114 views ### There is no simple group of order 36. I tried to do this as an exercise and wanted to ask if my proof is correct or if it is missing something. Thank you so much. Let G be a group such that \lvert G \rvert = 36 = 2^2 \cdot 3^2. Show ... 1 vote 0 answers 90 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 ... 1 vote 1 answer 55 views ### If the Sylow-2-subgroup of a group G is cyclic then can it surject \mathbb{Z}_2\oplus\mathbb{Z}_2 Suppose G is a group such that Sylow-2-subgroup is cyclic. Then can it surject into \mathbb{Z}_2\oplus\mathbb{Z}_2? If G is an abelian group and Sylow-2-subgroup is Cyclic, then it can not, by ... 4 votes 1 answer 194 views ### Let G be a group of order 7105, show that the order of the center of G is divisible by 35. Let G be a group of order 5\cdot 7^2 \cdot 29, show that the order of the center Z(G) of G is divisible by 35. Now, G contains only a 5-sylow, P_5, and since the conjugacy action of G/... 2 votes 0 answers 18 views ### Query regarding the order of an element in the centralizer. I was reading a text where they describe the structure of non-abelian order 12 groups, I quote a part of the text. "Let G be a non-abelian group of order 12. Let P_3 be a Sylow's 3-... 2 votes 1 answer 82 views ### If g: G \rightarrow S_4 is a group homomorphism with |\ker g|=24, then the group G is solvable. \DeclareMathOperator{\Ima}{Im} By the first isomorphism theorem of groups, we have that$$G/\ker g \cong g(G) \leq S_4$$Since the group$S_4$is solvable, we know that every subroup of$S_4$is ... 0 votes 0 answers 39 views ### How to find the element described in Sylow II I'm currently doing this question: Find two Sylow$3$-subgroups$H_{1}$and$H_{2}$of the alternating group$A_{4}$and an element$\sigma \in A_{4}$such that$H_{2}= \sigma H_1\sigma^{−1}$. I can ... 0 votes 0 answers 36 views ### A$2$-Sylow of$A_6$is self normalizing Let$A_6$be the alternating group of order$6$. Let$R$be a$2$-Sylow of$A_6$, I want to show that$R=N_{A_6}(R)$. What I have done is that we have that there is a monomorphism from$S_4$to$A_6$... 3 votes 1 answer 61 views ###$G\le S_n$in which all$g\ne e$have$<\sqrt{n}$cycles must be$\Bbb Z_p$or$\Bbb Z_p\rtimes\Bbb Z_q$Exercise 1.4.18 in Dixon & Mortimer's Permutation Groups is Let$G$be a permutation group of degree$n$, and suppose that each$x\ne1$in$G$has at most$k$cycles. If$n>k^2$, show that$G$... 2 votes 1 answer 76 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 (... 2 votes 1 answer 43 views ### Let$P\in{\rm Syl}_p(G)$and$K$a subgroup of$G$containing$N_G(P)$. Show that$N_G(K)=K$Question: Let$P\in{\rm Syl}_p(G)$and$K$a subgroup of$G$containing$N_G(P)$. Show that$N_G(K)=K$I'm wanting to use Frattini, but it looks like I need to use Frattini on$N_G(K)$. Certainly,$...
Source : Eliott Nicholson ( YT) , Group Theory Playlist, video n° 75, 22:40 [image below] Let $G$ be a group such that $|G|=p^{\alpha}m$ with $p$ a prime number and $m\in \mathbb N$. Let $\mathscr{C}$...