Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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1answer
57 views

On the maximum number of Sylow subgroups

I have some doubts regarding the first sentences below the fourth case in the more elementary solution by Prof. Samuel to problem 11856 of the Monthly. Here you have a screenshot of the relevant part ...
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1answer
47 views

Problems related to conjugacy classes and their sizes.

Is $Z( G_1 \oplus G_2 \oplus G_3 \oplus\cdots\oplus G_n) = Z(G_1) \oplus Z (G_2) \oplus Z(G_3) \oplus \cdots \oplus Z$ $(G_n)$ true, where $ G_1, G_2, G_3 \cdots G_n$ are finite groups?$Z$,here refers ...
3
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2answers
47 views

Subgroups of finite abelian groups.

For every subgroup $H$ of a finite abelian group $G,$ there exists a subgroup $N$ of $G$ such that $G/N \cong H.$ I need to prove this or give a counter example. I am aware of isomorphism theorems ...
5
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0answers
81 views

Almost all finite groups have order $2^n$?

This might be a stupid question, but here it goes: Is anything known about, whether: $$\lim_{n\to \infty} \frac{\#\{\text{Groups of order }2^n\}}{\#\{\text{Groups of order} \leq 2^n\}} = 1$$ (where ...
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0answers
27 views

Differences among the Group Cohomology with coefficients over a commutative ring and coefficients over an arbitrary $G$-module

Assume that $G$ is a finite group and $k$ is an arbitrary commutative ring. From the general theory we know that the group cohomology $H^{*}(G ; k)$ becomes a graded ring (with the cup product). ...
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3answers
55 views

Ley $G$ be a group of prime order $p$. Then $|Aut(G)|=p-1$

Let $G$ be a group of order $p$ where $p$ is a prime number( hence, $G$ is cyclic ) Prove that the group of automorphisms of $G$ has order $p-1$. Since $p$ is prime, for any homomorphism $\phi: G \to ...
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1answer
71 views

Troubles to understanding notation and some terminology on cohomology of finite groups

I am reading Adem, Milgram's book "Cohomology of finite groups" and I have some troubles with the notation. In particular, I don't get whether they are referring to the cohomology as a ring or as a ...
2
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1answer
36 views

Correlation amomg cohomology of a group $G$ and singular coohomology of its classifying space $BG$

Assume that $G$ is a finite group and let's denote by $BG$ its classifying space. Then we know that we can construct the cohomology $H^{*}(G ; M)$ with (local) coefficients, for a $G$-module $M$, in ...
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1answer
117 views

States of a Group Ring

Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Now taking the approach of orangeskid, consider the space $\mathbb{C}G$ as a Hilbert space with orthonormal basis $\delta^g$. $G$ acts on ...
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0answers
38 views

Orbits of the permutation action of a subgroup on its cosets

Consider a finite group $G$ and a subgroup $H \subseteq G$. There is a transitive group action of $G$ on the set of left cosets $gH$ by left multiplication, and the stabilizer of $gH$ is $gHg^{-1}$. ...
2
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1answer
72 views

If $G=\left<(12),(34),(45)\right>\subset S_5$, then $G\cong C_2\times S_3$

Let $G=\left<(12),(34),(45)\right>\subset S_5$. Show that $G\cong C_2\times S_3$. So my first idea was to set $a=(12)$, $b=(34)$ and $c=(45)$ and remark that $$G=\left<a,b,c\mid ab=ba,ac=ca, ...
4
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1answer
83 views

Number of subgroups equal to order of group

Here is a fun question. Consider the dihedral group $\mathcal{D}_4=\left\langle a,b\mid a^4=b^2=1, bab=a^{-1}\right\rangle$ of order $8$. This group has exactly $8$ genuine subgroups (but not all ...
3
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3answers
80 views

Can the dihedral groups be seen as subgroups of each other?

I am solving one difficult problem and now I need information on this: If $m\mid n$, is then possible to "embed" $D_m$ to $D_n$, or otherwise said, does $D_n$ have a subgroup isomorphic to $D_m$? ...
2
votes
2answers
79 views

Sylow subgroup of a product of subgroups

Let $UV$ be a subgroup of $G$ and $P$ is a Sylow $p$-subgroup of $G$. Suppose that $P \cap U \in \operatorname{Syl}_p(U)$ and $P \cap V \in \operatorname{Syl}_p(V)$. I need to show that $P\cap UV \in \...
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2answers
36 views

If $a$ has order $p$ and $aPa^{-1}=P$ then $a\in P$

If $G$ is a finite group, $P$ a $p$-Sylow subgroup of $G$, $a\in G$ has order $p$ and $aPa^{-1}=P$ then $a\in P$. This is proved in Rotman: Proof: We have $a\in N_G(P)$. If $a\notin P$ then $aP\in ...
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2answers
68 views

Supersolvable groups and sylow towers

Is it true that a supesolvable group has a sylow tower? How can I construct the tower kwowing than there's a principal serier with factors of prime order? Can anyone help me with an hint or an idea? ...
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1answer
25 views

Finitely generated Ideals of finite algebras

i would really appreciate any help with this question. So, the question is: How to prove that finitely generated ideals of finite algebras over the ring F are finite over F? Thanks.
4
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1answer
56 views

Let $\vert G \vert = p^n m$ where $m < 2p$. Show that $G$ has a normal subgroup of order $p^{n-1}$ or $p^n$

Let $G$ have order $p^n m$ where p is a prime and $p \nmid m$. Suppose $m < 2p$. Show that $G$ has a normal subgroup of order $p^{n-1}$ or $p^n$. I have tried to apply the Sylow Theorems but I ...
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0answers
75 views

Let $G$ be a finite group, $H$ be a $s$-permutable subgroup of $G$. if index $|G:H|=p$. then $H$ is normal subgroup of $G$.

$H$ is called $s$-permutable in $G$ if it permutes with every Sylow subgroup of $G$. Let $G$ be a finite group, $H$ be a $s$-permutable subgroup of $G$. if index $|G:H|=p$, where $p$ is an odd prime ...
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0answers
47 views

External Semidirect product and isomorphism

Let G and K be two groups and $\phi_1$ and $\phi_2: G \rightarrow Aut(K)$ be homomorphism. Q1: If $\phi_1$ not trivial homomorphism, can When can semidirect product of G and K using $\phi_1$ ...
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3answers
82 views

Index of intersection of two normal subgroups

Question: Let $G$ be a finite group and $H$ and $K$ are normal subgroup of $G$. If $[G:H]=2$ and $[G:K]=3$, determine $[G:H∩K]$. My attempt: I think $[G:H∩K] = 6$. Since $G$, $H$, $K$ are ...
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1answer
36 views

minimal subgroup is a direct product of simple groups

On page 112 of Dixon and Mortimer's Permutation Groups, Theoerm 4.3A (iii) says that every minimal normal subgroup $K$ of $G$ is a direct product $K=T_1 \times \cdots \times T_k$ where $T_i$ are ...
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3answers
82 views

Is there a non trivial normal subgroup of a group $G$, where $|G|=pm, \ \gcd(p,m)=1$?

In fact, the exercise I'm in is this: Suppose you have an irreducible polynomial $f(x)\in \mathbb{Q}[x]$ of degree $p$, where $p$ is a prime. Also suppose that $K$ is a splitting field of $f(x)$ over ...
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0answers
63 views

Smallest number $m$ with $gnu(m)=2017\ $?

$gnu(n)$ denotes the number of groups of order $n$ upto isomorphy. $moa(n)$ denotes the smallest number $m$ with $gnu(m)=n$ $$m=259,083,319,343,897,905=5\cdot 2011\cdot 24133\cdot 1067692187$$ ...
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1answer
57 views

Finding a 2-cocycle in $H^2(S_3, C_4)$

As far as I know, it holds $H^2(S_3, C_4)\cong C_2$ for the trivial operation of $C_4$ as a $S_3$-module. I have tried getting a $2$-cocycle (which is not a $2$-coboundary) by its defining equation: ...
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0answers
140 views

Is the group $G_{n,\phi} = \langle x_1 , \dots, x_n \mid x_i^2, (x_i x_j)^4, x_i x_{\phi(i)} x_{i+1} x_{\phi(i)} \rangle$ abelian?

I am working on a family of finitely presented groups and I asking me the following question. Let $\phi$ be an application from $\left\{1,\dots,n\right\}$ to $\left\{1,\dots,n\right\}$ (not necessary ...
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1answer
21 views

Normalizer of $H_0 \times H_m$ in $S_{2n}$

Let $H_0 \subsetneq \dotso \subsetneq H_m$ be a chain of subgroups of $S_n$ (the symmetric group of $n$ elements) such that holds: $N(H_i) = H_{i+1}$ and $N(H_m) = H_m$. I want to prove that the ...
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2answers
51 views

Prove that a normal subgroup $K$ is characteristic in a finite group $G$ if $\gcd(|K|, |G/K|) = 1$

This is a part of an exercise $2.2$ on a page $203$ in a book "Algebra: Chapter 0" by P.Aluffi. Let $K$ be a normal subgroup of a finite group $G$, and assume that $|K|$ and $|G/K|$ are relatively ...
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2answers
39 views

relation of classification of finite group and finite simple group.

I know that classification of finite simple group is completed. From the fact, can we say that classification of finite group is completed? I know a few relations of finite group and finite simple ...
3
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1answer
34 views

Which groups have only real representations?

An irreducible representation $\rho$ (with character $\chi$) of a finite group is called a "real" representation if its Frobenius-Schur indicator is 1: $$\frac{1}{\lvert G \rvert} \sum_{g \in G} \chi\...
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2answers
94 views

Let $G$ be a group and $N$ be a minimal normal subgroup of $G$, $G_p \in Syl_p(G)$, where $p$ is an odd prime divisor of $|G|$.

Let $G$ be a group and $N$ be a minimal normal subgroup of $G$, $G_p \in Syl_p(G)$, where $p$ is an odd prime divisor of $|G|$. Suppose that $M/N$ be a maximal subgroup of $G_pN/N$. then $M = PN$ for ...
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1answer
38 views

Order of product of non-disjoint cycles

Let $a$ and $b$ be two non-disjoint cycles of order $m$ and $n$. Is there any general formula for the order of $a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ...
3
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4answers
81 views

Is it true that the order of the group is a power of $2$ if every element has order $2$?

I read in this old question that If $G$ consists only of elements of order 2, then $|G|=2^m$ for some $m$. But it's not clear to me. I tested the base case $G=\{a,b,ab,e\}$ but induction ...
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0answers
22 views

Odd order subgroups of $PSL(2.q)$

Let q be an odd prime power. Is it true that every odd order subgroup of $PSL(2,q)$ is abelian ? If yes, how can it be proven ?
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1answer
26 views

If $G$ is p-nilpotent then $G$ has only one p-Sylow. Is it true?

Let be $G$ a group p-nilpotent. So $G$ has a p-normal complement $H$ that is a $p'$ Hall subgroup. I have read that if $G$ has a p-complement $H$ then this $H$ is unique. I don't understand: the p-...
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1answer
38 views

Centralizer of a Sylow $2-$subgroup of $PSL(2,q)$

Let q be an odd prime power. By a classic result, a Sylow 2−subgroup $P$ of $SL(2,q) $ is generalized quaternion. It is an irreducible subgroup of $GL(2,q)$ (since otherwise its natural ...
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2answers
37 views

$G$ nilpotent group and $N\trianglelefteq G$ then $[N,G]<N$, attempt of the proof

I need help in proving this fact: Let be $G$ nilpotent group and $N$ a normal and non trivial subgroup. Then $[N,G]$ is a proper subgroup of $N$. My attempt: I know the following fact: Let be $H$ ...
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1answer
15 views

Relation between the dual module and the dual of a vector space.

I need to know if there is a relation between the dual module of a subspace U of a finite dimensional vector space V, looked as a G-module, and the dual vector space of U. In this case, G is a finite ...
3
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1answer
42 views

Finite groups and prime divisors. Understanding how to deduce a claim from a certain proof.

In my algebra textbook, it goes like this. First, there is presented Cauchy's theorem: Let $G$ be a finite group, and let $p$ be a prime divisor of $|G|$. Then $G$ contains an element of order $p$....
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0answers
52 views

A Sylow $2-$subgroup of SL(2,q) is irreducible

Let $q$ an odd prime power. By a classic result, a Sylow $2-$subgroup of $SL(2,q)$ is generalized quaternion. How can I show that it is an (absolutely) irreducible subgroup of $GL(2,q)$ ? I have ...
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0answers
27 views

Permutation representation for low degree

Thanks for any answer. Suppose $n\leq 10$ and $n\neq 6$ and $k\geq 3$. How can I find all faithful permutation representation of $S_k$ in $S_n$? I mean is there any faithful representation except ...
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2answers
134 views

Conjugates and commutators for twisty puzzles — so what?

This question isn't just rhetorical. I want to know what I'm missing. Twisty puzzle tutorials keep talking about how useful conjugates (operation sequences of the form ${XYX}^{-1}$) and commutators ($...
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0answers
64 views

All but a finite number of finite simple groups are groups of matrices over $\mathbb{F}_q$

In the introduction to this honors thesis, http://people.math.gatech.edu/~jrabinoff6/papers/building.pdf I found this statement: Matrix groups defined over the finite fields $\mathbb{F}_q$ ...
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1answer
45 views

Order of subgroup generated by two cyclic subgroups in $S_6$.

Let $S_6$ be the symmetric group, and $\alpha=(13456)$ and $\beta=(132)$ be its two permutations. How can we find the order of the subgroup generated by $\alpha$ and $\beta$. SOl: $\alpha^5$=...
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0answers
32 views

Properties of non-abelian characters

I'm looking for some (short of) non-abelian generalization of the following result: Let $G$ be a finite abelian group and let $f$ be a function on $G$ with values in some field of characteristic ...
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1answer
26 views

Groups with order a product of unrelated distinct primes

Consider a group of size $n$, where $n$ is the product of distinct unrelated primes (two primes $p$ and $q$ are unrelated if $q \nmid (p-1)$ and $p \nmid (q-1)$). The claim is that there is only one ...
2
votes
1answer
61 views

If $|G/H|=4$ then $G$ is union of three proper subgroups

Let $G$ be a finite group and $N$ a normal subgroup of $G$ of index $4$. Prove that $G$ has a subgroup of index $2$ (hence normal) and if $G/H$ is not cyclic then $G$ is equal to the union of three ...
2
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0answers
62 views

Abelian groups of order 63

I am trying to learn abstract algebra on my own. Unfortunately I am confused and not sure how to proceed with the following question. I want to find all abelian groups of order 63. By theorem of ...
2
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1answer
70 views

“Concrete” Realisations of non-abelian finite groups

Many commutative groups could be imagined as something "concrete", for example $\mathbb N$ as an abstraction of operations on sets of objects, and from this $\mathbb Z, \mathbb Q$ and $\mathbb R$ are ...
2
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1answer
55 views

Proof of Lagrange's four square theorem using Cauchy-Davenport Theorem

Cauchy used the Cauchy-Davenport theorem to prove that $ax^2 + by^2 + c \equiv 0 \pmod p$ has solutions provided that $abc \neq 0$. Lagrange used this result to establish his four squares theorem. I ...