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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2
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1answer
116 views

representations of the dihedral group

Let $\rho_\epsilon(a)=\begin{bmatrix}\epsilon & 0\\0 & \epsilon^{-1}\end{bmatrix}$ and $\rho_\epsilon(b)=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$ I can prove that $\rho_\epsilon$ is ...
3
votes
0answers
78 views

Fusion of elements of order 8 in a finite simple group

Must a finite group $G$ with an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$ also have a normal subgroup of index 2? I ...
3
votes
2answers
225 views

Centralizer/Normalizer of abelian subgroup of a finite simple group

My concern is to look for a classification of : Centralizers/Normalizer of elementary abelian subgroups of a simple group. Centralizers/Normalizer of abelian subgroups(Not necessarily elementary) ...
4
votes
2answers
176 views

Exercise 5C10 in Isaacs' Finite Group Theory

Problem: Suppose that $G$ is simple group and has an abelian Sylow $2-$subgroup of order $8$. Show that the order of $G$ is divisible by $7$. Is there any hint to solve this problem? I'll be glad if ...
0
votes
1answer
27 views

irreps of $p^3$-group is faithful representation

Let $A$ be an irreps of $p^3$-group. Prove that $A$ is faithful representation. I know that $p^2$-group and $p$-group are abelian. I have to show, that $Ker A=e$ I have no idea how to start it
0
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1answer
56 views

How does Cauchy's theorem follow from Sylow's theorem?

Very quickly, Sylow's first theorem says a sylow p-subgroup of order $p^rm$ exists and Cauchy's theorem says if $p \vert |G|$ then there is an element of order $p$. It's often said that Cauchys ...
3
votes
1answer
85 views

Finite groups with unique minimal subgroup

Let $G$ be a finite group. Let $G$ has a unique non trivial minimal subgroup. Then $G$ is a p-group. How to prove the theorem which says that: If $G$ has a unique non trivial minimal subgroup and if ...
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4answers
105 views

Order of conjugate of an element given the order of this element

Let $G$ is a group and $a, b \in G$. If $a$ has order $6$, then the order of $bab^{-1}$ is... How to find this answer? Sorry for my bad question, but I need this for my study.
3
votes
1answer
91 views

If all the centralizers are finite, then does it follow that the group is finite?

As far as I have heard, centralizers play an important role in the theory of groups. My question arises from curiosity and the desire to understand how much control centralizers have over the group. ...
1
vote
4answers
77 views

How many isomorphism of $\phi :\mathbb Z_{4} \rightarrow \mathbb Z_{4}$?

I'm interested in how to find it, not the answer itself. I'm confuse to solve this question, I know isomorphism is bijective, and in this case it called Automorphism. But, I can't find a way how to ...
1
vote
1answer
72 views

If $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$.

Let $G$ be finite abelian group and $\hat G$ be its character group. I need hint proving that if $a\in G$ and $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$ (the identity element). I can prove it ...
1
vote
2answers
69 views

if $G$ is a group of order $p^n$ where $p$ is prime

If $G$ is a group of order $p^n$, where $p$ is prime and $n \geq 1$, prove that $G$ must have a subgroup of order $p$.
0
votes
1answer
30 views

$\phi : \mathbb{Z}_5 \to $ H is a homomorphism, where H is a 5 order group .

If $\phi(1) = a^3$, then $\phi(4)$ is ...? How to get the answer correctly, I'm still beginner in abstract algebra.
7
votes
1answer
184 views

Minimal counterexamples of the isomorphism problem for integral group rings

The isomorphism problem for integral group rings asks if two finite groups $G,H$ are isomorphic when their integral group rings $\mathbb{Z}[G]$, $\mathbb{Z}[H]$ are isomorphic. Quite a lot has been ...
0
votes
1answer
47 views

decompose a real representation of a group $C_2\times C_2

Let $\Phi$ be a real representation of the group $C_2\times C_2=\{e,a\} \times \{e,b\}$ such as $ \Phi(a)=\begin{bmatrix}5 & -4 & 0\\6 & -5 & 0 \\0 & 0 & 1\end{bmatrix}$ and $ ...
6
votes
1answer
108 views

Is there an infinite group that contains every finite group (and no infinite group) as a subgroup?

Question is in the title. For bonus points, construct the group $G$ such that it also has no infinite proper subgroups. (This second question relates to the Prüfer group, but that group is abelian, ...
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3answers
408 views

Given a group, how to show the distributive law and some examples (does the distributive law have to be an axiom?)

tl;dr: What properties does a set need for it to have the distributive law? Does it need to be an axiom? Given a group (i.e. satisfies the closure, associative, inverse and identity ...
2
votes
1answer
85 views

subgroups of finite cyclic group

Let $G=(g)$ be a finite cyclic group generated by $g$ with $|G|=n$, and let $d \in \mathbb{N}$ with $d|n$, then an unique subgroup $H$ of $G$ with $|H|=d$ exists. Proof of existence: $\exists m \in ...
2
votes
1answer
95 views

Connections and Differences between these Cayley Diagrams for $A_4$ and $S_4$ - Carter pp. 80, 82

Reference: Nathan Carter pp. 80, 82, ch. 5, Visual Group Theory Figure 5.24. As you will read in the next section, it is no coincidence that [the Cayley digram for $S_4$] looks cube-like. A Cayley ...
3
votes
1answer
47 views

$S_n$ acting on $\{1\;…\;n\}\times \{1\;…\;n\}$

Let $X=\{1,\;...\;n\}$ and $S_n$ act transitively on $X\times X$ i.e. $s:\;(m,n)\mapsto (s(m),s(n))$. Compute the orbits under this action. Attempt: I claim that there are only two orbits, ...
0
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2answers
54 views

On maximal subgroup's index of a finite group

Let $G$ be a finite group and $p$ be a prime. Assume that for every maximal subgroup $M$ of $G$ we have $[G:M]$ is not $\equiv1$ mod $p$. Prove that $G$ has a normal Sylow $p$-subgroup. ...
2
votes
0answers
225 views

Maximal subgroups of direct product of solvable groups

Let $G_1$ and $G_2$ be finite solvable groups and $M$ be a maximal subgroup of $G=G_1\times G_2$ show that one of the following holds: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
0
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2answers
58 views

If the product $x_1x_2…x_n=1$, prove that each $x_i=1$.

Let $G$ be a finite solvable group and let $x_1,x_2,...,x_n$ be elements of $G$ of pairwise relatively prime orders. If the product $x_1x_2...x_n=1$, prove that each $x_i=1$. I have no idea. Tell me ...
3
votes
1answer
66 views

Let $G$ be a finite group and let $H$ be a solvable subgroup of $G$.

Let $G$ be a finite group and let $H$ be a solvable subgroup of $G$. Let $N=N_G(H)$ and assume that $N/H$ is a nonabelian simple group. Prove that $N=N_G(N)$. $N=N_G(N)$. That means ...
3
votes
1answer
85 views

Number of subgroups of order 48 in $\mathbb{Z}_{8} \oplus\mathbb{Z}_{12}$

I thought it would have sufficed to show that every subgroup of G=$\mathbb{Z}_{8} \oplus\mathbb{Z}_{12}$ must be formed by couples (a,b) whose set of a's and the set b's form a subgroup of ...
2
votes
1answer
86 views

Let $H$ be a subgroup of $G$ which contains $N_G(P)$ for some Sylow $p$-subgroup $P$ of $G$.

Let $G$ be a finite group and $p$ be a prime. Let $H$ be a subgroup of $G$ which contains $N_G(P)$ for some Sylow $p$-subgroup $P$ of $G$. Suppose $P \subseteq H^g$ for some $g \in G$. Prove ...
2
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0answers
36 views

Is this dual-spaced norm based on $L_2$ norm

I am reading the book of Claes Johnson about Numerical Solution of Partial Differential Equations by the Finite Element Method and particularly pages 34 and 98. I wrote these notes to my craft Is ...
2
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1answer
106 views

A question about perfect group

Let $G$ be a finite group. Show that if $G = G'$, then $Z$$\left( G/{Z\left( G \right)} \right)=1$. My attemp is here. Fact: $G' \le N$ if the quotient group $G/N$ is abelian. Since $G = G'$, by the ...
5
votes
1answer
139 views

Prove that if $A \vartriangleleft G$ is abelian, then $A$ has a complement in $G$.

Let $G$ be a finite group. Suppose that the intersection of all of the maximal subgroups of $G$ is trivial. Prove that if $A \vartriangleleft G$ is abelian, then there exist $U \subseteq G$ such that ...
3
votes
1answer
224 views

Let $G$ be a finite group. Suppose $N,H,K \subset G$ are subgroups such that $NH=G$ and $(N \cap H)K=G$. Prove that $N(K \cap H)=G$. [closed]

Let $G$ be a finite group. Suppose $N,H,K \subset G$ are subgroups such that $NH=G$ and $(N \cap H)K=G$. Prove that $N(K \cap H)=G$. I have no idea. Give me some hints.
4
votes
2answers
176 views

An exercise about simple groups

Let $G$ be a finite group, $D=\{(g,g):g\in G\}$ is a subgroup of the direct product $G\times G$. Show that $G$ is simple if and only if $D$ is a maximal subgroup of $G\times G$. I tried to prove by ...
3
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2answers
103 views

The Quaternion Group and other Group Theory Counterexamples [closed]

The quaternion group $Q_8$ is a common counterexample to many statements. For example, even though every subgroup is normal, it is not abelian, a direct product, or even a semidirect product! In ...
3
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1answer
105 views

A finite group which has exactly eight Sylow $7$-subgroups

Let $G$ be a finite group which has exactly eight Sylow $7$-subgroups. Prove that there exist a normal subgroup $N$ of $G$ such that its index is divisible by $56$ but not by $49$. Give me some ...
0
votes
2answers
237 views

quotient group Z/3Z equated to 0,1,2

This a followup question on the first answer to this post: Why the term and the concept of quotient group? In the first answer, Lahtonen says that for the quotient group Z/10Z, one can "equate 9 ...
7
votes
4answers
205 views

Let $G$ be a finite group which has a total of no more than five subgroups. Prove that $G$ is abelian.

Let $G$ be a finite group which has a total of no more than five subgroups. Prove that $G$ is abelian. I can prove that if $\left|G\right|\leq5 $ then $G$ is abelian. Is it equivalent to this ...
1
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1answer
121 views

There exist two distinct elements of $G\setminus H$ which commute?

Let $(G, *)$ be a finite group of odd Because $G$ has odd order, every element x frhas order greater than 2, so I think, the condition that H is non-commutative is just to confuse.
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1answer
39 views

Solving for $x$ where $p^x \equiv 1 \pmod {q\#}$

For a given primorial $q\#$, you can generate a subset of the reduced residue system by using the power of a prime $p$ where $p > q$. For example, for $5\#$, we can use the powers of $7$ to ...
2
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0answers
63 views

Finite group with elementary abelian centralizer

Let $G$ be group of order $q(q^{2}-1)/2$ (where $q=p^n$ is an odd prime power) such that $C_{G}(P)$ is elementary abelian for every Sylow $p$-subgroup $P$. Is there any classifications of this type ...
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2answers
223 views

Let $G$ be a finite group and $p$ be a prime. Suppose that every proper subgroup of $G$ of order divisible by $p$ is a $p$-group.

Let $G$ be a finite group and $p$ be a prime. Suppose that every proper subgroup of $G$ of order divisible by $p$ is a $p$-group. Prove that every two distinct Sylow $p$-subgroup of $G$ intersect ...
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vote
2answers
147 views

Finite group with the property that every element of order a power of $p$ is contained in a conjugacy class of size a power of $p$

Let $p$ be a prime and let $G$ be a finite group with the property (*) that every element of order a power of $p$ is contained in a conjugacy class of size a power of $p$. i) If ...
3
votes
3answers
53 views

irrep of a non unit element in the finite group

Let $G$ be a finite group. Prove following statement. $\forall g \in G$ such that $g \neq e$ $ \exists$ a complex irrep $\rho$ such that $\rho(g) \neq E$ I have no idea how to start it. I can prove ...
1
vote
1answer
57 views

Factoring over prime fields

Suppose I have two numbers in Fp that are multiplied together: r*s Is there anything special that needs to be done when prime factorizing since this is a finite field (i.e., is this what is referred ...
0
votes
1answer
75 views

Finite group that has order not divisible by 3

If $o(G)=m$ and $m$ is not divisible by $3$ then show that for every element $x\in G$ there exists another element $y\in G$ such that $x = y^3$.
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1answer
42 views

number of complex irreps

Can irreducible complex representation of a finite group of be exhausted by a) two 1-dimensional and two fifth-dimensional representations? b) five one-dimensional and 1 five-dimensional? My ...
1
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1answer
115 views

Nonabelian group of order $p^4$ [closed]

Let $P$ be a nonabelian group of order $p^4$, where $p$ is a prime, and let $A$ be a subgroup of $P$ maximal with the property of being normal and abelian. Prove that $A$ is of order $p^3$. Thanks a ...
1
vote
1answer
39 views

Qustion about field and sub-group

$F$ is a field and and $H$ is finite sub-group of $(F,\cdot)$ ($F$ without the $0_F$). I need to prove that $H$ is cyclic. I can use this fact - Can we conclude that this group is cyclic?. (I don't ...
4
votes
2answers
158 views

A finite group with the property that all of its proper subgroups are abelian

Let $G$ be a finite group with the property that all of its proper subgroups are abelian. Let $N$ be a normal subgroup of $G$. Prove that either $N$ is contained in the center of $G$ or else $G$ has a ...
4
votes
1answer
250 views

Group of order $1575$ having a normal sylow $3$ subgroup is abelian.

Question is to prove that : If a group $G$ with $|G|=1575=3^2\cdot5^2\cdot 7$ has a normal sylow $3$ subgroup then : sylow $5$ subgroup is normal sylow $7$ subgroup is normal In this ...
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vote
2answers
61 views

How to apply Thompson's A×B lemma to show this nice feature of characteristic p groups?

How does one apply Thompson's A×B lemma (Lemma 24.2 on page 112 of Aschbacher's Finite Group Theory) in order to prove this nice lemma (Lemma 31.16 on page 160)? In the book, I basically don't ...
2
votes
1answer
150 views

Use of the commutator to reduce the degree of a permutation

I've found the following claim here (page 4, in the proof of thm 1.1). Set $A$ the support of a permutation $\tau$ in $S_n$, with $\deg\tau=|A|=k$. Given a permutation $\pi$ in $S_n$ such that ...