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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Construction of Permutation Group (bounded order) from a Set of Permutation

Given a set of permutation $S$. It has $|S|$ (The cardinality of $S$) elements. Consider a subset $A\subset S$. We can construct a group using $A$. One possible algorithm could be- We start ...
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52 views

The number of Sylow $5$-subgroups in $S_6$

Find the number of sylow $5$-subgroups in $S_6$. First: $ord(S_6)=6!=2^4\cdot 3^2\cdot 5=144\cdot 5$, so $n_5|144$ and $n_5\equiv 1\pmod5$, where $n_5$ is the number of sylow $5$-subgroups. Since ...
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45 views

Group of order $210$

Is there a $G$ group of order $210$ satisfy following properties: for some $x,y \in G$, order of $x$ is $5$, order of $y$ is $3$ $x^4y^2=(yx)^6$.
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Is there a name for the class of infinite groups with no proper infinite quotients?

As the title suggests, I would like to know if there is there a name for the class of infinite groups with no proper infinite quotients? I came across this paper http://onlinelibrary.wiley.com/doi/10....
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37 views

$U(st)$ is isomorphic to $U(s)\oplus U(t)$ where $s$ and $t$ are relatively prime.

Suppose $s$ and $t$ are relatively prime.Show that $U(st)$ is isomorphic to $U(s)\oplus U(t)$. I want to show that $\phi$ :$U(st)$ $\rightarrow$ $U(s)\oplus U(t)$ defined by $\phi(x)=(x\bmod s,x\bmod ...
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Faithful irreducible character and Sylow subgroup

I am trying to solve the (very nice) exercise 5.25 from Isaacs, character theory. Assume that every Sylow subgroup of $G$ has a faithful irreducible character. Show that $G$ has one also. The ...
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Suppose G is a group of order 4 and $x^2=e$ for all $x$ in G. Prove that G is isomorphic to $Z_2\oplus Z_2$. [duplicate]

Suppose $G$ is a group of order $4$ and $x^2=e$ for all $x$ in $G$. Prove that $G$ is isomorphic to $Z_2\oplus Z_2$. My attempt: (1) Show that $G$ is abelian. \begin{align*} \text{Take }x,y\in G.&...
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34 views

Question on abelian group, does $G/H$ abelian $\iff [G,G]\leq H$.

Let $G$ a group and $H\lhd G$ a normal subgroup. I have a theorem that say that if $G/H$ is abelian, then $[G,G]\leq H$. I wondered if the converse hold, does it ? I recall that $[G,G]=\left<ghg^{-...
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Character and centralizer of a subgroup

I am trying to solve the exercise 2.15 from Isaacs Character Theory. Let $\chi\in Irr(G)$ be faithful and let $H$ be a non-trivial proper subgroup of $G$ such that $\chi_{H} \in Irr(H)$. Show that $...
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35 views

The Rotation Group of a Cube

Show that the group of rotations of a cube is isomorphic to $S_4.$This proof is from Gallian's Abstact Algebra Theorem $7.3$ Proof:Using the Orbit-Stabilizer Theorem we know that the group of ...
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63 views

Twisty Puzzle Solving Program

I'm writing a program to help me solve a twisty puzzle. In this case it's the face-turning octahedron. I'm representing the puzzle as a group with face twists as generators. The facelets are in a list ...
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40 views

Maximum order of element in group of units in a ring

Let $s$ be a natural number and $U(s)$ be a group of units in the ring $\mathbb{Z}/s\mathbb{Z}$. Let $\phi(s) = 2^{k_1}p^{k_2}$, where $p$ is a an odd prime number. I don't understand why the maximum ...
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When will the classification of a property of a group be complete

I am studying pronormal Hall $\pi$-subgroups of a finite group $G$ All Hall $\pi$-subgroups of a finite solvable groups are known to be pronormal as this follows from Hall's theorem Recently, it has ...
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24 views

Suppose n is even positive integer and $H$ is a subgroup of $\mathbb Z_n$ [duplicate]

Suppose $n$ is even positive integer and $H$ is a subgroup of $\mathbb Z_n$. Prove that either every member of $H$ is even or exactly half of members of $H$ are even. Same question have been asked ...
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Is $\mathbb{Z}_4 \times \mathbb{Z}_ {12} \times \mathbb{Z}_9$ cyclic?

I find on internet this: $\mathbb{Z}_m \times \mathbb{Z}_n$ is cyclic if and only if $\gcd(m,n)=1$. Then I do the next steps: $\gcd(4,12,9)$ is 1. Then I assume that $\mathbb{Z}_4 \times \...
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42 views

Is there a simple solution to these 2 equations without trying all possible values?

Now I have two equations and the computation is in a finite field GF(p), where p is a prime. $x, y$ are unknown, and $a, b$ are known. ($0<y<p-1$, and $0<ax<p-1.$) $\begin{cases} x^y =...
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Proving that the index of a subgroup of $S_n$ that keeps a specific set invariant has a certain order

Let $n \in \mathbb{N}, n ≥ 2$, and $k \in \{1, 2, ..., n-1\}$, and let $A \subseteq \{1, 2, ..., n\}$ with $|A| = k$. Furthermore, let $G$ be a subgroup of $S_n$ that fixes $A$, i.e. for all $π \in G$,...
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Question on character theory.

Let $\phi :G \to \text{Aut}(V_0)$ be the regular representation of $G$. Define a representation $\Theta$ of $G \times G$ on $V_0$ by $\Theta(g_1,g_2)e_g=e_{g_1gg^{-1}_2}$, i.e it maps the element $(...
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38 views

If a finite group $G$ has a normal subgroup $N$ of order $p$ where $p$ is the smallest prime divisor of the group order, then $Z(G)$ is nontrivial

Let $G$ be a finite group and $p$ the smallest prime number with $p \mid |G|$. I want to show: if $G$ has a normal subgroup $N \trianglelefteq G$ of order $p$, then $G$ has a nontrivial center. My ...
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26 views

Determining the size of a subgroup of H

Given that $|GL_{2}(Z_{5})|=480$ find the index of $H$ in $GL_2(\mathbb{Z}_5)$. $$ H=\begin{pmatrix} a & b \\ 0 & c \\ \end{pmatrix} \\a,c \neq 0 , \\a,b,c\in \mathbb{Z}_5 $$ I proved in ...
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Counting homomorphisms of groups

How many homomorphisms are there from $A_{5} × S_{3} × A_{4}$ to $\mathbb Z/2\mathbb Z$? I know that $Z/2Z$ is generated by ome element and of order two. I know the respective order of the rest of ...
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30 views

Isomorphism between finite groups with specific property is unique

I am looking for a clever justification for a statement I believe to be true. I would like to show that given an isomorphism, $\Phi$, between two finite groups, if it is known that $\Phi(a) = b$, then ...
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1answer
22 views

$|\hom(G, \Bbb Z/2)|>1 \iff |G:[G,G]|$ is even

Let $G$ be a finite group. My problem is: $$|\hom(G, \Bbb Z/2)|>1 \iff |G:[G,G]| \mbox{ is even}.$$ I know that $\hom(G, \Bbb Z/2)=\hom(G^{ab}, \Bbb Z/2)$, where $G^{ab}:=G/[G,G]$. If $|\hom(G^{ab}...
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1answer
42 views

Character theory - exercise 5.16 from Isaacs

Hi I am trying to solve the following exercise. Let $H$ be maximal subgroup of a finite group $G$ and let $\chi=(1_H)^G$. Let $\psi$ be a non-principal irreducible constituent of $\chi$. Then $Ker \...
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Character theory - exercise 3.4 from Isaacs

Let $G$ be a simple group and suppose $\chi\in Irr(G)$ with $\chi(1)=p$ a prime, Show that a Sylow $p-$subgroup of $G$ has order $p$. The indication provided is: if the Sylow $p-$subgroup $P$ is ...
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Prove that $\alpha$ is an automorphism of $Z_n$.

Let $ r \in U(n)$[We define $U(n)$ to be the set of all positive integers less than $n$ and relatively prime to n]. Prove that the mapping $ \alpha : Z_n \rightarrow Z_n $ defined by $ \alpha(s)=sr$ ...
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21 views

Show that a set is a right transversal for $S_3$ in $S_4$

I am given the set A ={e,(14),(24),(34)} I'm supposed to show that this set is a right transversal for S3 in S4, meaning that every right coset of S3 contains exactly one element of A. I'm getting ...
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Number of conjugacy classes or bounds for the index of stabilizers on finite Coxeter groups.

For a finite group $G$ denote by $\operatorname{conj}\left(G\right)$ the number of conjugacy classes of $G$ and let $q\left(G\right)$ be the quotient: \begin{align} q\left(G\right)=\frac{\...
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What is the smallest number $n$ with $gnu(n)=2016$?

$gnu(n)$ denotes the number of groups of order $n$ upto isomorphy. What is the smallest number $n$ with $gnu(n)=2016$ ? The numbers upto $n=2047$ do not satisfy the given property. I do not know ...
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1answer
52 views

$p$-groups have normal subgroups of each order [duplicate]

Suppose $|G|=p^n$. Then $G$ has a normal subgroup of order $p^m$ for every $0\le m\le n$. By induction. It is clearly true for $n=0$. Now suppose $k<n$ and $H_i$ is a normal subgroup of $G$ of ...
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Explicit computation of $H^2(\mathbb{F}_p^n, \mathbb{R}/\mathbb{Z})$.

I'm interested in the computation of the second cohomology group of the elementary abelian group $\mathbb{F}_p^n$ with coefficients in $\mathbb{R}/\mathbb{Z}$: $$H^2(\mathbb{F}_p^n, \mathbb{R}/\...
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1answer
51 views

Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$

I want to determine how many subgroups does the additive group $G:=\Bbb Z_{14} \oplus \Bbb Z_{6}$ have? There are many related posts in our site, for instances: here and there. However, it ...
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Chief factors and local formation

Every thing below is concerned with finite groups. My question is about this paper A class of groups is a collection $\mathcal{X}$ of groups with the property that if $G \in \mathcal{X}$ and if $H \...
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Show that $S_5$ does not have a quotient group isomorphic to $S_4$

Show that $S_5$ does not have a quotient group isomorphic to $S_4$. If we to assume that $H$ is such a group, than $H$ must be normal in $S_5$ and $|H|=|S_5|/|S_4|=5$. So $H$ must be isomorphic to $\...
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Question about finite abelian group

Let G be an abelian group of order $mn$ where $\gcd(m,n)=1$. I proved that $mG$ and $nG$ are subgroups and that $G=mG+nG$ and now i want to prove the three things: the sum is direct, i.e. $mG\cap ...
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31 views

About the conjugacy classes of a finite group

Let $K_1, \cdots , K_n$ denote the conjugacy classes of a finite group $G$. For $x \in K_s$, define $n_{ijs} = |\{(y, z) \in K_i × K_j : yz = x\}|$. I want to show that $n_{ijs} = n_{jis}$. How ...
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Automorphism Group of a cyclic p-group

I want to show that the automorphism group of $C_p^{k}$ is cyclic for an odd prime $p$. I know that the order of $Aut(C_n)$ is $\phi(n)$ and so the order of $C_{p^{k}}$ is $\phi(p^{k}) = p^{k-1}(p-1)...
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Show that all elements of $\left<a,b,c\right>$ are of the form $a^ib^jc^k$ (comprehension)

Let $G=\left<a,b,c\right>$ a subgroup of $\mathfrak S_6$ where $a=(123),b=(456)$ and $c=(23)(45)$. Show that every element of $G$ can be uniquely written as $a^ib^jc-k$ where $0\leq i,j\leq 2$ ...
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Looking at two versions of Fundamental Abelian groups theorem

$G$ is a finite abelian group. Then it can be expressed with a direct product of cylic groups with prime power order $G$ is a finite abelian group with order $n$. Then it can be expressed as ...
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Do there exist simple groups of order $n(n+1)$ for some integer $n>1$?

In the comments to this question, it was remarked that the question of whether a finite group of order $p(p+1)$ is necessarily not simple becomes fairly interesting if we do not assume that $p$ is a ...
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1answer
31 views

Why does Lagrange come into play here? [duplicate]

I am given this problem, and the solution reasons differently from me and obscurely, to me Let $G$ have order $30$. Show that it is not simple. The solution is apparently So, what I don't ...
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Sylow theorem; Sylow $III$

Just curious, I have't seen any statement that answers this so far. And I'm not at the stage of being able to or given a proof of the theorem so I am not sure. Sylow $III$; If $G$ has order $n=p^...
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Are the normalizers of Sylow p-subgroups isomorphic?

Let $G$ be a finite group and $A,B \in \text{Syl}_p G$, for some prime $p$. Is it always true that the normalizers $N_G(A)\cong N_G(B)$? I just need a hint to get started, because I don't know where ...
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1answer
91 views

Group of order $p^2+p $ is not simple [duplicate]

Can some one please give me a hint to prove that every group of order $p^2+p$ is not simple?
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1answer
22 views

first homology group with coefficients in divisible group

I had (perhaps very elementary) doubt in the understanding of the computation of first homology group of a finite group over a divisible group. Let $\pi$ be a finite group of order $n$ and $D$ be a ...
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The point of a group-theoretic Chinese Remainder Theorem?

It states that for coprime $m,n$ nonzero integers, $C_{mn} \cong C_m \times C_n$. However, I know a theorem that says Cyclic groups with the same order are isomorphic. So $C_{mn} \cong C_m \...
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36 views

Non-abelian groups of order $50$

Are there non-abelian groups of order $50$? If so, how many elements of order $2$ can they have? Such a group would have a unique $5$-Sylow-group, so it would have at least $25$ elements not of order $...
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1answer
19 views

Crossed homomorphism from semi-direct product: confusion in definition

(Ref: this) Let $\pi \times_{\varphi} G$ be semi-direct product in which $G$ is normal and $\pi$ is complement. Let $\omega$ be another complement of $G$ in above semi-direct product (so $\pi \...
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Ordering of elements in the base of a group

In section 4.6.7 of HANDBOOK OF COMPUTATIONAL GROUP THEORY, the authors use an ordering $\prec$ for the elements in a coset. That ordering, $\prec$, was defined in section 4.6 as follows. Throughout ...
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33 views

Website or book with Hasse diagrams of subgroups

I need to look at Hasse diagrams of very many groups, especially high powers of small symmetric groups. Is there any place where I could look them up? Calculating them myself would be a huge amount of ...