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Questions tagged [finite-groups]

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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Compute all the irreducible representations of $\mathbb{Z}_{2}^{k}$

Define $\mathbb{Z}_{2}^{k}=\underbrace{(\mathbb{Z}/2\mathbb{Z})\times(\mathbb{Z}/2\mathbb{Z})\times...\times(\mathbb{Z}/2\mathbb{Z})}_{k\text{ times}}$. I understand that the irreducible ...
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$p\equiv3\pmod 4$ if $p$ divides the number of automorphisms [duplicate]

Let $p$ be an odd prime and $G$ a group with $p+1$ elements. If $p$ divides the number of automorphisms of $G$, prove that $p\equiv3\pmod 4$. Can somebody help me, please? I have no idea how to ...
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Verify $(\chi_1\mid \chi_2)=0$, $(\chi_1\mid \chi_1)=1$, $(\chi_2\mid \chi_2)=1$

Let $\rho_{1}(k)=\begin{bmatrix}e^{2\pi ik/n}\end{bmatrix}$ and $\rho_{2}(k)=\begin{bmatrix}e^{4\pi ik/n}\end{bmatrix}$ where $k=0,1,2,...,n-1$ and $n\geq3$, where $\rho_1$ and $\rho_2$ are ...
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0answers
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Describe all the irreducible representations of $G=\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}$

If $G=\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z},$ then the irreducible representations of $G$ are given by \begin{align*}\rho_{j_{1}j_{2}}(k_{1},k_{2}) =\begin{bmatrix}e^{2\pi i(j_{1}k_{1}/n)}...
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About a Corollary of Isaacs' “character theory of finite groups”: does the converse implication hold?

Let $N \lhd G$ (with $G$ finite) and let $\chi \in \mathrm{Irr}(G)$ be such that $\chi_N=\theta \in \mathrm{Irr}(N)$. Then the characters $\beta \cdot \chi$ for $\beta \in \mathrm{Irr}(G/N)$ are ...
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56 views

Group of order $5^k\cdot 8$ has normal subgroups of order $5^{k},5^{k}\cdot2,5^{k}\cdot4$

Let $G$ be a group of order $5^k\cdot 8$. I was trying to prove that there are normal subgroups of order $5^{k},5^{k}\cdot2,5^{k}\cdot4$. I saw the following statement: Let $P$ be a $p$-Sylow ...
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Do the cosets of in invariant (normal) subgroup consist entirely of classes?

Let $H$ be an invariant subgroup of some group $G$. Is it then true that the cosets of $H$ consist entirely of conjugacy classes? The question was prompted by something I was thinking about in ...
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For distinct primes $p$ and $q$, does any group of order $p^2q$ have a subgroup of order $p$ (without using Sylow or Cauchy)?

The Details: I'm reading "Contemporary Abstract Algebra (Eighth Edition)," by Gallian. This is based on Exercise 7.40 of the "Cosets and Lagrange's Theorem" section ibid. Here it is for convenience: ...
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66 views

A single set of moves $S$ that, if repeated, solves the Rubik's cube from any state

I am looking for a proof verification. I often find these concepts simple, but struggle to communicate them clearly. Communication in mathematics is very important to me: Examples could be: Any ...
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1answer
25 views

$p$-group problem

Let $A,B,C$ are three subgroups in a way that $1<A \triangleleft B \triangleleft C$. With $B/A$ and $C/B$ are $p$-groups. Then prove that $|C|$ is also a $p$-group. I have been trying to prove it ...
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Does a list of elements' orders in an abelian group determine the group?

Let $G=\{a_1,a_2,\ldots,a_n\}$ be an abelian group and let $m_1\leq m_2\leq \ldots \leq m_n$ be the list of orders of all the elements of $G$. E.g., for the Klein 4-group $V$ we get the list $1,2,2,2,...
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38 views

How to compute the number of equivalence classes under the relation $a\sim b\iff a=x^m b x^n$?

Let $G$ be a finite group. Fix an element $x\in G$, and denote by $\sim$ the equivalence relation on $G$ given by $a\sim b \iff \exists m,n\text{ such that }a=x^m b x^n$. Example: Let $G=\langle(123),...
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6answers
274 views

Subgroup of a finite group $H$ and $G$

I regret to admit that this has been confusing me for much longer than I would like. I just can't wrap my head around some parts of this question and I'd like some guidance. Let $G$ be a finite ...
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1answer
47 views

Does $D_4$ have a verbal subgroup of order 4?

Does $D_4$ have a verbal subgroup of order 4? How did this question arise: In the comments $Q_8$ ad $D_4$ were pointed to be a possible counterexample to this question: Is it true, that for any two ...
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1answer
73 views

Torsion-free Abelian Groups of Finite Rank and Free Groups (Fuchs) - Self study

I want to solve the following problem (Fuchs, "Infinite Abelian Groups", Vol.$2$, pp. $153$ Ex. $4$): "Let $A$ be a torsion-free group of finite rank $n$ and $F$, $F'$ free subgroups of $A$ of rank $...
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25 views

Number of elements in a multiplicative group of $ (complex)^{2^n} $ th roots of unity for a specific n.

The number of elements of the multiplicative group $ (complex)^n $ th roots of unity is n .But for a fixed n the number of elements in a multiplicative group of $ (complex)^{2^n}$ th roots of unity ...
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Lift $O(\mathbb{Z}/p\mathbb{Z})$ to “something” in $O(\mathbb{Z}/p^2\mathbb{Z})$

So the question was from a result of Serre which basically says if $H$ is a closed subgroup of $Sp_{2n}(\mathbb{Z}_p)$ that maps surjectively onto $Sp_{2n}(\mathbb{Z}/p\mathbb{Z})$, then $H=Sp_{2n}(\...
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1answer
35 views

Find $3$ non-isomorphic groups of order $2012$

Find $3$ non-isomorphic groups of order $2012$. Is the following correct? First of all, we have the two non-isomorphic abelian groups $\mathbb Z_{2012}$ and $\mathbb Z_{2}\times\mathbb Z_{1006}$. ...
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1answer
36 views

How to prove this property of this group of order $20$ without the Sylow theorems?

In Artin's Algebra under the section on the Class Equation is the exercise The class equation of a group $G$ is $1+4+5+5+5$. (a) Does $G$ have a subgroup of order $5$? If so, is it normal? (b) Does ...
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3answers
43 views

Proof that a group is finite

$(G,.)$ is a group and $H$ a subgroup of it so that $G-H$ is finite. Prove that G is finite. I found the next proof: $f_a:H \to G-H ,f_a(x)=ax, a\in G-H$ . Since $f_a$ is injective and $G-H$ is ...
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Does there exist some sort of classification of finite verbally simple groups?

Let’s call a group verbally simple if it does not have any non-trivial verbal subgroup. Does there exist some sort of classification of finite verbally simple groups? $G^n$, with $G$ being a finite ...
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1answer
29 views

Problem with subgroups and sets [on hold]

I had the next problem at a contest at my school today: Let $(G,•)$ be a group and $H,K$ $2$ subgroups in $G$. For all two nonempty sets $A,B$ in $G$, it is noted $A•B=\{a•b\space|\space a \in A,b \in ...
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1answer
29 views

Characters of $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 $

Follow up question to $\mathbb{Z}_2 \oplus \mathbb{Z}_2$, how would I find the characters of $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$? The Cayley table for this group: \begin{align*} \...
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1answer
66 views

Subsets of $\mathbb Z/n\mathbb Z$ disjoint with some of its shifts

Are there any descriptions of all subsets $X$ of $\mathbb Z/n\mathbb Z$ with the following property: there exists $a\ne 0$ in $\mathbb Z/n\mathbb Z$ such that $X$ is disjoint with $X + a = \{x + a \...
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1answer
34 views

Subsets of $\mathbb Z/n\mathbb Z$ that remain disjoint with themselves under shifts

Are there any descriptions of all subsets $X$ of $\mathbb Z/n\mathbb Z$ such that for any $a\ne 0$ in $\mathbb Z/n\mathbb Z$, $X$ is disjoint with $X + a = \{x + a \pmod n\mid x \in X\}$?
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Is it true, that for any two non-isomorphic finite groups $G$ and $H$ there exists such a group word $w$, that $|V_w(G)| \neq |V_w(H)|$?

Is it true, that for any two non-isomorphic finite groups $G$ and $H$ there exists such a group word $w$, that $|V_w(G)| \neq |V_w(H)|$? Here $V_w(G)$ stands for the verbal subgroup of $H$, generated ...
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1answer
41 views

Characters of $\mathbb{Z}_2 \oplus \mathbb{Z}_2$

From the Cayley table: \begin{align*} \begin{array}{c | c c c c } & (0,0) & (0,1) & (1,0) & (1,1)\\ \hline (0,0) & (0,0) & (0,1) & (1,0) & (1,1)\\ (0,1) &...
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35 views

Questions Related to the Monster Group Size

Assume the term $\phi$-reduction refers to recursive application of Euler's totient function $\phi(n)$ as follows until the result $1$ is reached. $\quad\phi_0(n)=n$ $\quad\phi_1(n)=\phi(n)$ $\quad\...
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2answers
37 views

Explicit example of a direct sum over two simple groups

I want to understand the notion of a direct sum properly. I learn best via examples so it would very helpful if I could get a concrete example using two simple groups. for examples suppose we want to ...
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1answer
33 views

Show that both side of the Plancherel formula are linear in $f$.

Plancherel Formula: Let $f$ and $h$ be functions on G. Then, $$\sum_{s\in G}f(s^{-1})h(s)=\frac{1}{|G|}\sum_{i}d_{i}Tr(\hat{f}(\rho_{i})\hat{h}(\rho_{i}))$$. I understand $f$ is linear if $f_1+f_2(x)=...
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0answers
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Calculate the character degrees of a finite group $G$.

Let $G$ be a finite group and $K$ be a group of order $8$. Suppose that $G/K\cong M_{12}$ where $M_{12}$ is one of the Mathieu group. QUESTION: How to calculate the all complex character degrees ...
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0answers
41 views

Elementary proof that centre of finite $p$ group is not trivial [duplicate]

Prove that the centre of finite $p$ group is not trivial. I have found on many links proofs of this property, but all of them use the "Class Equation". I would like to know if there is a proof which ...
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3answers
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How do I know that $\mathbb{Z}_{175}$ is not an additional subgroup of order $175$?

Here was the original problem statement. Enumerate all non-isomorphic groups of order $175$. See that $|G| = 175 = 5^2\cdot7$. Therefore by Sylow's first $H \leq G$ & $|H| = 25$ in addition to ...
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0answers
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Quick proof that $SL_2(\mathbb Z/n\mathbb Z)\cong \oplus_{p\mid n}SL_2(\mathbb Z/p^{e_p}\mathbb Z)$

I'm looking for a quick proof that $SL_2(\mathbb Z/n\mathbb Z)\cong \oplus_{p\mid n}SL_2(\mathbb Z/p^{e_p}\mathbb Z)$ for some nonnegative integers $e_p$. I've argued as follows, but I'm hoping for a '...
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1answer
31 views

Construct a group that has exactly 5 elements of order 4.

Construct a group that has exactly 5 elements of order 4. I wonder if it is possible. I tried $U(8)$ but it has $\{[1], [3], [5],[7]\}$ as elements which has order $4$ but it has only $4$ elements. ...
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2answers
83 views

Is the normality of a subgroup dependent on which group is its parent?

It is important to understand the relationship of normal subgroups to their parent. One concept that needs to be understood is whether the normality of a subgroup does not depend on which parent group ...
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0answers
27 views

Question about commutativity of conjugacy classes

I have an exercise in group theory which asks to show that conjugacy classes commute even though the group is not abelian, i.e. that $C_1C_2 = C_2C_1$ for any classes $C_1$ and $C_2$. The proof given ...
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1answer
29 views

Suppose $G$ is a group of order $n$ . Let $p$ and $q$ be distinct primes which divide $n$. Can we say that $G$ has a subgroup of order $p\cdot q$?

Suppose $G$ is a group of order $n$ . Let $p$ and $q$ be distinct primes both of which divide $n$. Can we say that $G$ has a subgroup of order $p\cdot q$?
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1answer
98 views

Is a finite centerless metabelian group always a semidirect product of two abelian groups?

Suppose $G$ is a finite centerless metabelian group. Is it true that it is a semidirect product of two abelian groups? It does not seem true to me, but I failed to find any counterexamples. Actually, ...
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0answers
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How do I turn a group presentation into a multiplication table?

Suppose I have a group presentation and I know for a fact that this is a presentation of some finite group. How do I create a multiplication table from this presentation?
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If $G = H \cup K \cup L$, then $[G:H] = [G:K] = [G:L]=2$ [duplicate]

This is a problem from Isaacs's Algebra. Let $G$ be a finite group and $H, K, L$ be proper subgroups of $G$ such that $G= H \cup K \cup L$. He asks us to show that, in this case, $[G:H] = [G:K] = [G:L]...
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0answers
35 views

For $n>1$, let $H$ be the set of all products in $S_n$ of a multiple of four transpositions. Show $H=A_n$.

I'm reading "Contemporary Abstract Algebra (Eighth Edition)," by Gallian. This is Exercise 5.80 ibid. Answers that use only tools available in the textbook so far are preferred. For $n>1$, let $...
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1answer
97 views

Is every group of order 117 abelian?

Let $G$ be a Group of order 117. Then is $G$ abelian? Note that $117 = 13*3^2$ and that $13 \equiv 1 \bmod 3$, so A group of order $p^2q$ will be abelian doesn't help.
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1answer
34 views

For which $n\in \Bbb N$ is $H_n:=\{\alpha^2\mid \alpha\in S_n\}\cong A_n?$

I'm reading "Contemporary Abstract Algebra," by Gallian. This is inspired by Exercise 5.73 and Exercise 5.74 ibid. I have a preference for answers using only the tools available in the textbook so ...
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vote
3answers
42 views

Cauchy's theorem proof clarification (group theory)

Cauchy's theorem says that if $G$ is a finite group with $p | |G|$ (when $p$ is prime), then $G$ contains an element of order $p$. When following the proof from wikipedia: https://en.wikipedia.org/...
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1answer
76 views

Proving $\left(\mathbb{Z}/p^{d} \mathbb{Z}\right)^{\times}$ is cyclic for prime p

My assignment asks me to prove $\left(\mathbb{Z}/p^{d} \mathbb{Z}\right)^{\times}$ is cyclic for prime $p>2$ and for any positive integer $d$. They propose proving this by induction. The base ...
6
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2answers
76 views

Can any finite group of order $n$ be embedded in $SL_n(\mathbb{Z})$?

If $G$ is a finite group of order $n$, it can be embedded in $S_n$ by Cayley's theorem and therefore in $GL_n(\mathbb{Z})$ and $SL_{n+1}(\mathbb{Z})$. Can $G$ be embedded in $SL_n(\mathbb{Z})$?
4
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1answer
32 views

Maximal subgroup of Wreathed 2-groups

Definition A $2$-group $S$ is called wreathed if it is isomorphic to $(C\times C)\rtimes \langle i \rangle$ where $C$ is a cyclic group of order $2^n$ and $i$ is an involution with action $(a,b)^i=(...
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0answers
19 views

Show $\nexists x\in S_7$ with $x^2=(1234)$ but there is at least two $x\in S_7$ with $x^3=(1234)$. [duplicate]

I'm reading "Contemporary Abstract Algebra," by Gallian. This is Exercise 5.48 ibid. and I want to answer the question using the tools available in the textbook so far. (A free copy of the book is ...
1
vote
1answer
32 views

Group of order $pqr$

Assume $G$ is a group of order $pqr$, with $p, q, r$ distinct primes. Let $P, Q, R$ be their corresponding Sylow subgroups. In addition, assume $P\subseteq C(G)$ and $R\subseteq N(Q)$ where $C$ and $N$...