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
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End of the proof of Burnsides $p^aq^b$ Theorem

Theorem. Every group of order $p^aq^b$ ($p,q$ primes $a,b \geq 0$) is soluble. Near the end of the proof of Burnsides $p^aq^b$ Theorem, we want to prove the following If $\rho:G ...
0
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0answers
39 views

Middle bit of the proof of Burnsides $p^aq^b$ Theorem

Theorem. Every group of order $p^aq^b$ ($p,q$ primes $a,b \geq 0$) is soluble. Part of the proof needed to prove the above theorem is where you prove that: $\displaystyle \frac{\chi(C)}{\chi(e)}$ ...
1
vote
1answer
38 views

Number of orthonormal sets of vectors $\leq$ dim of the vector space

Theorem. If $V,W$ are arbitrary representations of $G$, say $$V=V_1^{a_1}\oplus \dots \oplus V_k^{a_k} $$ $$W=V_1^{b_1}\oplus \dots \oplus V_k^{b_k} $$ Then, $$\langle \chi_V,\chi_W \rangle ...
1
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2answers
45 views

Start of proof of Burnsides $p^aq^b$ Theorem

Theorem. Every group of order $p^aq^b$ ($p,q$ primes $a,b \geq 0$) is soluble. Proof. Enough to prove that no non-abelian simple groups have order $p^aq^b$. [Then break $G$ into simple pieces ...
0
votes
1answer
49 views

left and right multiplication for Cayley graphs

Correct me if i am wrong but i have written down the following: If $X$ is a finite group, with subset $S$ and corresponding Cayley graph $G$ The edge set for a Cayley graph is defined such that two ...
2
votes
1answer
64 views

Where can I find Wielandt's original proof of Sylow's Theorem?

I have seen several proofs of Sylow's Theorem based on Wielandt's method. Everyone gives credit to Wielandt's proof of Sylow's theorem, but ironically everyone puts their own spin on it. Where can I ...
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2answers
32 views

Generator of group, find the inverse, solve equation

Given the prime number $p=101$ Find a generator of the group $\mathbb{Z}_p^{\star}$. How many generators of $\mathbb{Z}_p^{\star}$ are there? Find $5$ generators. Find the inverse of $\overline{83}$ ...
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0answers
38 views

Subgroups and Direct Products of $\pi$-closed groups are also $\pi$-closed.

Let $K$ be a class of finite groups, then this class is called closed iff i) homomorphic images of groups in $\mathcal K$ lie in $\mathcal K$, ii) subgroups of groups from $\mathcal K$ lie in ...
1
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1answer
49 views

Defining a ring homomorphism in proving $|C| \cdot \frac{\chi(C)}{\dim V} $

Lemma irreducible representation $\rho: G \rightarrow GL(V)$, $C$ conjugacy class, then $$|C| \cdot \frac{\chi(C)}{\dim V} $$ is an algebraic integer. In the start of this proof we have: ...
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2answers
49 views

Describe all extensions of the identity map of $\mathbb{Q}$ to an isomorphism mapping $\mathbb{Q}(\sqrt[3]{2},\sqrt{3})$

Here is the full question : Describe all extensions of the identity map of $\mathbb{Q}$ to an isomorphism mapping $\mathbb{Q}(\sqrt[3]{2},\sqrt{3})$ onto a subfield of the algebraic closure of ...
2
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1answer
68 views

Groups of Order $210$

By the "$2n$-test", proving that a group of order $210$ cannot be simple. Is there another way to prove this? Would you use Sylow Theory?
3
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1answer
30 views

Is there any generalization of Frobenius group?

Let $G$ act on $\Omega$ transitively and $\chi(g)$ is equal to number of the elements fixed by $g$. If $\chi(g)\leq 1$ for all $g\in G\setminus\{e\}$ then $G$ is a Frobenius group. There are many ...
10
votes
2answers
93 views

How “big” can the center of a finite perfect group be?

A perfect group is a group where the derived (commutator) subgroup $G'$ of $G$ equals $G$. $G'$ measures the "non-abelian-ness" of $G$, in a sense. This suggests that "many" elements of perfect $G$ ...
2
votes
3answers
48 views

Prove that $H_n=\{\sigma \in S_n \mid \sigma (i) \equiv i \pmod 3 \}$ with $n≥2$ is a subgroup of $S_n$.

Prove that $H_n=\{\sigma \in S_n \mid \sigma (i) \equiv i \pmod 3 \}$ with $n≥2$ is a subgroup of $S_n$. I'm doing this problem and I don't know if my approach is correctly done. First of ...
3
votes
2answers
116 views

Does this define a Ring?

I am working on some rings practice questions and I have come across one where I am having a bit of difficulty with checking some of the ring axioms: Let $G$ be a finite group, let $ C(G) = \{ f: ...
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0answers
41 views

Problem about finite group whose proper subgroups are abelian. [duplicate]

Let $G$ be a finite group with the property that all of its proper subgroups are abelian. Let $N$ be a proper normal subgroup of $G$. Show that either $N$ is contained in the center of $G$ or else $G$ ...
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0answers
35 views

Subgroups of the dihedral group D_n modulo Aut(D_n)

This question is related to this math.se question. Consider the dihedral group $D_n = \langle r,s \rangle.$ Two subgroups $G, H \leq D_n$ are said to be ''isomorphic'' if there is an $f \in ...
3
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0answers
35 views

Irreducibility of a certain polynomial associated to an irreducible representation of a finite group

Let $k$ be an agebraically closed field of characteristic 0. Let $G$ be a finite group of order $n$. A representation of $G$ is a homomorphism $\psi: G \rightarrow GL(V)$ where $GL(V)$ is the general ...
0
votes
2answers
95 views

A finite abelian group has order $p^n$, where $p$ is prime, if and only if the order of every element of $G$ is a power of $p$

Suppose that G is a finite Abelian group. Prove that G has order $p^n$, where p is prime, if and only if the order of every element of G is a power of p. I tried the following route, but got stuck. ...
2
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0answers
30 views

If $C_A(F(A)) \le F(A)$ and $C_B(F(B)) \le F(B)$, then this also holds for $AB$ if $A,B \unlhd G$.

Let $A, B \unlhd G$ be normal subgroups of a finite group $G$ such that $$ C_A(F(A)) \le F(A) ~\mbox{ and }~ C_B(F(B)) \le F(B). $$ where $F(G)$ denotes the Fitting Subgroup of $G$. I want to show ...
2
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1answer
28 views

Finding subgroups of $G=\displaystyle\normalsize{\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}}\LARGE_{/}\large_{\langle(1,0)\rangle}$

I'm doing this exercise: Find all the subgroups of $G=\displaystyle\normalsize{\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}}\LARGE_{/}\large_{\langle(1,0)\rangle}$ This is my try: ...
4
votes
1answer
81 views

Existence of non-abelian group of order n

I know this is an old quetion, but I've certainly been disappointed with the given answers. The question is: There exists a characterization of the natural numbers $n$ such that there exist at least ...
2
votes
1answer
39 views

The minimal normal subgroups of a maximal subgroup $L$ if two minimal normal subgroups of $G$ are not in $L$

If a finite group $G$ contains a maximal subgroup $L$ and two minimal normal subgroups not in $L$, then every minimal normal subgroup of $L$ is contained in the subgroup generated by the minimal ...
1
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1answer
48 views

Simple method to determine the sign of the permutation $x \rightarrow x^{-1}$ on a finite group

Let $G$ be a finite group. Let $f: G\rightarrow G$ be the map defined by $f(x) = x^{-1}$. Is there a simple method to determine the sign of the permutation $f$? The motivation is as follows(I ...
3
votes
1answer
38 views

What could be said about centralizers of normal subgroups if $G$ contains a simple, non-abelian maximal subgroup

Let $G$ be a finite group containing a maximal simple and non-abelian group, is it true that the centraliser of each normal subgroup is either trivial or a minimal normal subgroups? EDIT: Maybe to ...
0
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1answer
69 views

If the commutator subgroup is abelian, is it necessary trivial? [closed]

Let $G$ be a group. We define the commutator of $a$, $b$ in $G$ as $[a,b]:=aba^{-1}b^{-1}$. Let $C=\langle[a,b] \mid a,b\in G \rangle$ be the commutator subgroup of $G$. Suppose that $C$ is abelian. ...
1
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1answer
21 views

Congruence Class of Negative Integers in a Multplicative Group

As part of a larger problem, I need to find the subgroup of $(\mathbb{Z}/7\mathbb{Z})^*$ generated but the congruence class of -1. I understand that this is a multiplicative group with elements {1, ...
6
votes
1answer
181 views

Proving that a group of order $pqr$ (with conditions on those primes) is abelian.

I'm doing this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Also suppose that $p \not\equiv 1$ (mod $r$), $p \not\equiv ...
4
votes
1answer
34 views

All maximal subgroups are complement

Let $G$ be a finite group such that for any maximal subgroup $M$ and a subgroup of $H$, we have $MH=G$ or $MH=M$. Can we say something about this group ? Note that the equality is claerly satisfied ...
1
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1answer
34 views

If two quotient groups are semi-simple, then a third build from both is semi-simple too.

I call a group semi-simple if it is the direct product of non-abelian simple groups. Let $G$ be a finite group and let $M, N \unlhd G$ such that $G/N$ and $G/M$ are both semi-simple. Prove that ...
3
votes
1answer
50 views

For a subgroup $H$ of a finite group $G$ , when does $|Aut(H)|$ divides $|Aut(G)|$?

Let $H$ be a subgroup of a finite group $G$, then is it true that $|Aut(H)|$ divides $|Aut(G)|$? What if we also assume $G$ is abelian? (I know that $|Aut(H)| \space \big| \space |Aut(G)|$ if $G$ is ...
0
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2answers
47 views

Non-generator elements of a group and Intersection of all maximal subgroups.

A non-generator element $u$ of a group $G$ is defined as, If $H\not=G,$ then $\langle H,u\rangle\not=G$ for any $H\le G.$ Show that set of all non-generators of $G$ is a subgroup of the ...
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0answers
15 views

Finite subgroups of $O_4(\mathbb{Q})$

I have a problem with the classification of finite subgroups(up to isomorphism) of $O_4(\mathbb{Q})$ (or $GL_4(\mathbb{Z})$). I know about classification of $GL_2(\mathbb{Q})$. Maybe somebody knows ...
0
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1answer
121 views

Let $G$ be a group. Let $G/K$ be an abelian group. Prove that $C=\{e\}$.

I'm stuck at this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Also suppose that $p \not\equiv 1$ (mod $r$), $p ...
0
votes
3answers
32 views

direct product of cyclic and non-cyclic group together.

consider direct product of two finite groups, one is cyclic and the other one is not, is the direct product cyclic? if both groups are not cyclic,what we can say about direct product of them? I ...
0
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2answers
93 views

How are quaternions a finite set?

I'm having trouble understanding how Quaternions are a finite set when you can express a quaternion as Q = a + ib + jc+ kd, because a, b, c, d are $\in$ of $\Re$ would this not mean that the set is ...
0
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2answers
64 views

Group Theory: Showing that a subgroup is isomorphic to a product of groups

I have the following question, where the topic being tested is cosets, order and Lagrange's theorem: Suppose that every element $x$ in a group $G$ satisfies $x^2 = e$. Prove that $G $is abelian. ...
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1answer
23 views

Problem in permutation groups involving conjugates

I have to find a permutation $a$ satisfying $ a xa^{-1}=y$ where $ x=(12) (34)$ and $y=(56) (13)$ My attempt in solving the problem was- $$ a(12)(34)a^{-1}= a(12)(a^{-1}a)(34)a^{-1}= ...
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1answer
37 views

Automorphism group of a non_abelian p_group

Let G be a non abelian p_group. When is set of all automorphisms group of G a p_group?
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1answer
40 views

Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup.

I'm stuck on this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup $P$. What ...
2
votes
1answer
89 views

Is $A_{4}\times Z_2\simeq \langle g,h \mid g^{12},h^2,{gh}^{12}, gh=hg\rangle$?

Is $A_{4}\times Z_2\simeq \langle g,h \mid g^{12},h^2,(gh)^{12}, gh=hg\rangle$? In addition, is $\operatorname{Aut}(A_{4}\times Z_2)= \operatorname{Aut}(A_{4})\times ...
9
votes
1answer
133 views

How many different groups of order $15$ there are?

I wanted to share with you my resolution of this exercise. How many different groups of order $15$ there are? My resolution: We're looking for groups such that $|G|=15=3\cdot 5$. Then: $G$ ...
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1answer
37 views

Conditions for Nilpotency of inverse image of homomorphism.

Let $\varphi : G \to L$ be a homomorphism and $U \le L$. Under what conditions is $\varphi^{-1}(U)$ nilpotent, if $U$ is nilpotent? And a closely related question. If $UN/N$ is a nilpotent subgroup ...
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0answers
17 views

Example such that $HN/N ~\mbox{char}~ G/N$ and $N~\mbox{char}~G$, but $H$ not characteristic in $G$

If $H/N$ is characteristic in $G/N$ and $N$ is characteristic in $G$, then $H$ is characteristic in $G$, a proof could be found here or here. The notation, i.e. speaking about subgroups $H/N$ implies ...
2
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1answer
77 views

How to compute the pointwise stabilizer subgroup of a fixed-point subspace?

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
1
vote
1answer
51 views

Finite groups acting on strings.

Let $s = abcdandsoon.. \ \in \Sigma^*$. Let $|s| = n$ be the length of $s$. Consider all permutations of the positioned symbols that make up $s$, such that $s$ is fixed under the permutation. So if ...
2
votes
2answers
64 views

Question on Proof that $O_p(C/(C\cap F(G)) = 1$ for $C = C_G(F(G))$.

I have a question on the proof of a lemma about the Fitting subgroup, I mention all used facts: If $N \unlhd G$ and $A ~\mbox{char}~ G$ be a characteristic subgroup of $G$. Then i) $A$ is normal in ...
0
votes
2answers
34 views

cyclic group contain normal subgroup of prime index

Let $G$ be finite cyclic goup i wont to show that $G$ contain normal subgroup of prime index. A group G is cyclic if $G$=$ \langle a \rangle$, for some a$\in$$G$. A finite cyclic group of order n ...
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0answers
38 views

Question on proof that maximal normal abelian subgroup is self-centralising in nilpotent groups

The following is known about finite groups: (*) If $G/Z(G)$ is cyclic, then $G$ is abelian. Proposition: Let $G$ be a nilpotent finite group and $N$ a maximal abelian subgroup of $G$. Then $C_G(N) = ...
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2answers
36 views

What are the transitive groups of degree $4$?

How can I find all of the transitive groups of degree $4$ (i.e. the subgroups $H$ of $S_4$, such that for every $1 \leq i, j \leq 4$ there is $\sigma \in H$, such that $\sigma(i) = j$)? I know that ...