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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.

Sylow's theorems give information about the numbers of subgroups of fixed order a finite group contains.

Often used with and .

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Sylow theorem and tetrahedron

According to Sylow's theorem, the group of isomorphism (rotations and mirror symmetries) of tetrahedron has a subgroup of order 8. How does one find it？ Moreover, is there a method to find a Sylow p-subgroup, or a subgroup of any order, of any…
Charlie Chang
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Prove that any group of order 392 has a normal group of order 7 or 49

So far I have tried the following: By Sylow's theorem there must be 1 or 8 $Syl_7 (G)$. If there is only 1, we are done since this would be the normal subgroup of size 49. So suppose there are 8. Let G act on the set of 8 by conjugation this…
Math Lady
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Struggle on Sylow p-subgroup

I'm in trouble on Dummit and Foote Abstract Algebra Ex. 6.2 13: Let $P,Q$ be distinct Sylow p-group with maximal $|P \cap Q|$. Show $N_G(P \cap Q)$ cotains more than one Sylow p-group and each pair intersect in $P \cap Q$. It's easy to showing…
Maddy
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Prove that a group $G$ of order $375$ has a subgroup of order $15.$

Prove that a group $G$ of order $375$ has a subgroup of order $15.$ My thought: $5|O(G)\implies G$ has a subgroup $H$ of order $5.$ $3|O(G)\implies G$ has a subgroup $K$ of order $3.$ $|H\cap K|=(e)$ and so $|HK|=\dfrac{|H|.|K|}{|H\cap K|}=15$ Now…
Sriti Mallick
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$N_G (P \cap Q) $ has more than one Sylow $p$-subgroup under some conditions.

Let $p$ be a prime number. Let $G$ be a group with more than one Sylow $p$-subgroup Over all pairs of distinct Sylow $p$-subgroups, let $P$ and $Q$ be chosen so that $|P \cap Q|$ is maximal. I want to prove that $N_G(P\cap Q)$ has more than one…
Seongqjini
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Let $F$ be a free centralizer in $G$. Then $F=P\times A$, where $P$ is a Sylow $p$-subgroup of $F$ and $A$ is abelian.

This question is from the Proposition A.23.2 in page 520 of the book ``Berkovich, Yakov, and Zvonimir Janko. Berkovich, Yakov; Janko, Zvonimir: Groups of Prime Power Order. Vol. 2. Walter de Gruyter, 2008.'' Suppose $G$ is a finite nontrivial…
bfhaha
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How many sylow-p subgroups are there of A5

Calculate the number of Sylow p-subgroups of $A_5$ We have $|G|=60=2^2⋅3⋅5$ Let $n_p$ be the number of Sylow p-subgroups of G. By Sylow's third theorem, we have $n_3∈{1,4,10}$. But G contains 20 elements of order 3, each of which generates a Sylow…
Alex Carter
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A p subgroup of a group is contained in a sylow p subgroup

I am trying to solve a problem involving automorphism of a group.There needs the following argument: a p subgroup is contained in a sylow p subgroup.Is it true?I can't prove it,may be it is elementary.Plz help me. I tried it producting with a sylow…
Panja
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Sylow Subgroup is

G be a finite group.P is a Sylow p-subgroup which is contained in the center of G.Show that there is a normal subgroup N of G such that G=PN.(Herstein problem,page 103 prob 16) Give some idea.P is normal in G,but can't proceed further. Sorry I…
Via
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Element of prime power in normalizer

Let $K$ be Sylow $p$-subgroup of a finite group $G$; than prove that if $x \in N_G(K)$ and the order of $x$ is power of $p$, than $x \in K$. My try , Let $K$ be a Sylow $p$-subgroup, $o(K) = p^k$ and since $x$ is in $N_G(K)$ then $ xKx^-1 = K$. Now…
Tutankhamun
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Show that the center of a group of order 60 cannot have order 4?

This seems to be related to Sylow's theorems, but I have no idea how?
user2277550
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Sylow subgroup related

In a group of all invertible $4 \times 4$ matrices with enteries in the field of $3$ elements, What is the cardinality of any $3$- sylow subgroup?
user344521
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intersection of all $2$-sylow in $S_n$

I am currently stuck on an exercise that asks to determine explicitly the subgroup $O_2(S_n)
pozio
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Sylow p-subgroup: Understanding a proof

I don't understand the part of the followng proof that I underlined with red. I don't see why $PN/N \cong P/(P \cap N)$ implies $PN/N$ is a p-subgroup of $G/N$.
Twnk
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How many 3-Sylow groups are in a group of order 126?

Let $G$ be a group of order $|G| = 126 = 2 \cdot 3^2 \cdot 7$. Let $H \leq G$ be a subgroup of order $|H| = 14$ and $\varphi:G \rightarrow H$ be a surjective group homomorphism. How many 3-Sylow groups are in $G$? (Let $s_3$ be the number of 3-Sylow…
Martin Thoma
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