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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17 views

How can we show that the closure property holds in the character group of a finite abelian group?

Let $G$ be a finite abelian group of order $n$. Let $f_1, f_2, \dots, f_n$ be the characters of $G$ where $f_1$ is the principal character and others are non-principal characters. So we now have a set ...
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3answers
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Prove the two preimage sets have the size under a surjective group homomorphism (for a finite group)

So I have a finite group $G$ and a surjective group homomorphism $\phi: G \to G'$. I was asked to show that for any $b,c\in G'$, $|\phi^{-1}(\{b\})|=|\phi^{-1}(\{c\})|$. As a hint, it told me to use ...
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165 views

Can a finite group act transitively on an infinite set?

I had a question on my algebra exam, asking me to show that in case of a transitive action of a finite group $G$ on a set $X$, $|X|$ divides $|G|$. In case of a finite set, this is trivial, so I tried ...
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16 views

An example to illustrate a complex-valued function $f$ that is a character of a finite group $G$

I am learning this theorem: If a complex-valued function $f$ is a character of a finite group $G$ (i.e. $f$ has the multiplicative property $f(ab) = f(a)f(b)$) with identity element $e$, then $f(e) = ...
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Given a finite solvable group $G$, prove that a minimal normal subgroup $H$ is a $p$-group

Given a finite solvable group $G$, and a minimal normal subgroup $H$, prove that $H$ is a $p$-subgroup for some prime $p$. My Attempt: I am trying to write this proof without using the term "...
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1answer
44 views

Classification of the Finite Abelian Groups

I am studying "Abstract Algebra" by Gregory T. Lee. I am going through chapter 5, "Direct Products and the Classification of Finite Abelian Groups" at the moment, but it has been ...
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31 views

Automorphism group of $C_{8}\times C_{2}$

I am trying to figure how to compute $Aut(C_2 \times C_8)$. Using GAP the answer is $Aut(C_2 \times C_8)\simeq C_2\times D_8$. I know that $Aut(C_8)\simeq C_2 \times C_2$ and that $Aut(C_2) \simeq 1 $,...
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32 views

What we need to show, $ab=1\bmod k$ or $(ab)\bmod n=1\bmod k$? For proving $U_k(n)≤U(n)$

I need to show that, For each divisor $k$ of $n$, $U_k(n)$ is subgroup of $U(n)$ where, $U_k(n)=\{x\in U(n) : x=1\bmod k\}$ My attempt: as $U(n)$ is finite group for each $n\in\mathbb{Z}^+$. Hence ...
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37 views

Explicit image of $A_5$ inside $PGL(2,\mathbb{C})$

The list of finite subgroups of $PGL(2,\mathbb{C})$ is well known. The cyclic subgroups are easily written down. I was wondering if it is possible to explicitly write down a finite subgroup which is ...
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31 views

Order of elements of the group of $3 \times 3$ upper-triangular matrices over $\mathbb{Z}/p\mathbb{Z}$ with $1$'s in the diagonal

Suppose $p$ is an odd prime. Let $G$ be the group of $3 \times 3$ upper-triangular matrices over $\mathbb{Z}/p\mathbb{Z}$ with $1$'s down the diagonal. Show that every element of $G$ has order that ...
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1answer
47 views

Growth rate of finite simple groups [closed]

Let $S$ be a finite generator of a finite simple group $G$. Then the word length of $g \in G$ where $g = \prod_{k=1}^n s^{\pm i_k}_k$ is defined as $i_1 + \cdots +i_k$. Now $\gamma(n)$ is the ...
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2answers
52 views

$\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p\times \mathbb{Z}_p$ [duplicate]

I try to prove if de groups $\mathbb{Z}_{p^{2}}$ and $\mathbb{Z}_p\times \mathbb{Z}_p$ are isomorphic. I was using the fact that $ \mathbb{Z}_ {mn} $ is isomorphic to $ \mathbb{Z}_m \times \mathbb{Z}...
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46 views

Intersection of group $\mathbb{Z}_p \times \mathbb{Z}_p$ and $\mathbb{Z}_{p^2}$ [closed]

Consider the finite groups, $\mathbb{Z}_p \times \mathbb{Z}_p$ and $\mathbb{Z}_{p^2}$ for prime $p$. What is the intersection of these two groups? My naive guess is $\mathbb{Z}_p \subset \mathbb{Z}_p \...
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69 views

How to show that the $f$ is one-one and onto?

Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $n = |xy|.$ My attempt : Since every element of $G$ can be ...
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1answer
78 views

Intersection of subgroups of a finite abelian group

Let $G$ be a finite abelian group and let $g \in G$ be a non-trivial element. I want to show that the intersection over all subgroups $G'$ of $G$ such that $G/G'$ is cyclic and $g \not \in G'$ is ...
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2answers
51 views

$|G:H|$ and $|H|$ are coprime [closed]

Suppose $G$ is a finite group with a normal subgroup, $K$, and another subgroup, $H$, such that $[G:H]$ and $|H|$ are coprime. Is it true that $[HK:H]$ will divide both $[G:H]$ and $|H|$? Why is this?
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1answer
48 views

Why does the dicyclic group have exactly one involution?

This is Exercise 12.2(a) of Roman's "Fundamentals of Group Theory: An Advanced Approach". According to this search, it is new to MSE. The Details: Roman defines, on page 350 ibid., the ...
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36 views

Why $\gcd(12,30)$ equals the number of homomorphisms from $\Bbb Z_{12}$ to $\Bbb Z_{30}$? [duplicate]

The following is a snippet from Gallian's Contemporary Abstract Algebra: Chapter 10-Example 10: We determine all homomorphisms from $\Bbb Z_{12}$ to $\Bbb Z_{30}$. By $\phi(g^n) = (\phi(g))^n$ for ...
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58 views

Probability that position $i$ is a peak in $\sigma$, where $\sigma$ be a uniformly random permutation of $\{1,\ldots,n\}$

Let $\sigma$ be a uniformly random permutation of $\{1,\ldots,n\}$. That is $\sigma(1),\sigma(2),\ldots, \sigma(n)$ is a permutation and it is chosen uniformly from one of the $n!$ permutations. ...
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1answer
35 views

Multiplication/Cayley tables for the Dihedral Groups

I am currently doing a group theory problem, which asks for the multiplication table of the dihedral group $D_4$. Having looked up the answer online, I do not understand how some of the elements arose....
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50 views

Subgroups of finite simple groups.

Let $G$ be a nontrivial finite group and $X_G$ the set of all the proper subgroups of $G$, $X_G:=\{H\subseteq G\mid H\le G \wedge H\ne G\}$. Lagrange's theorem put a limitation on the order of the ...
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24 views

factor group $\mathbb{Z}^{*}_{175} /\mathbb{Z}_{2}$

Suppose I have a factor group $\mathbb{Z}^{*}_{175} /\mathbb{Z}_{2}$. What is it? $$\mathbb{Z}^{*}_{175}=\mathbb{Z}^*_{25}\times \mathbb{Z}^*_{7}=\mathbb{Z}_{20}\times \mathbb{Z}_{6}.$$ $(\mathbb{Z}_{...
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1answer
40 views

What is the decomposition of $𝕂[G/H]$ in terms of irreducible representations?

Let $G$ be a finite group and $𝕂$ be a field. Any $G$-set can be linearized to give a $𝕂$-representation of $G$. Each $G$-set is decomposed into a coproduct of indecomposable (transitive) $G$-sets $...
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How can a cyclic group generated by 1 be a proper subgroup of a field?

Niederreiter and Winterhof, Applied Number Theory, page 26, Theorem 1.4.12 begins: Theorem 1.4.12$\;$ If $F$ is a finite field, then the order of $F$ is a prime power $p^r$, where the prime number $p$...
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111 views

Show that two finite two generator groups are isomorphic

I was faced by a question that I can't solve. Any help would be great! Let $A$ and $B$ be groups with the following properties: \begin{cases} |A| = 9 \cdot 3=27\\ A = \left<a,b\right> \\ a^{...
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84 views

Prove that if $G$ is a finite group with an even class number, then $G$ is of even order. Give an example that the converse fails.

Prove that if $G$ is a finite group with an even class number (the number of conjugacy classes of a group $G$ is called its class number) then $G$ is of even order. Give an example that the converse ...
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1answer
37 views

Real characters of odd degree

Suppose that $G$ is a finite group and $\chi$ is an irreducible real character, namely that $\chi(g) \in \mathbb{R}$ for every $g \in G$. Is it true that if $\chi(1)$ is an odd number, then $\chi$ is ...
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53 views

On the cardinality of the subgroup of central automorphisms

Let $p$ be an odd prime and $G$ be a purely non-abelian p-group of nilpotency class 2 in which $G/G'$ is an elementary abelian p-group of rank $n$. Let $Aut_{c}(G)$ be the subgroup of central ...
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40 views

Let $A\le B$ and $B\le C$. What are the possibilities of ${\rm ord}(B)$ if ${\rm ord}(A)=60$ and ${\rm ord}(C)=4200?$

I had a nice problem in my abstract algebra homework, and when I wanted to go deeper and generalize it, I got a little bit stuck. Let $A$ be a subgroup in $B$ and $B$ be a subgroup in $C$. Is there ...
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27 views

Character triples isomorphism and real characters

A character triple is a triple of the form $(G,N,\theta)$ where $G$ is a finite group, $N$ is normal in $G$, $\theta \in Irr(N)$ and $\theta$ is $G$-invariant. For the concept of character triple ...
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18 views

Obtaining the product of any element with a Hamiltonian path in a Cayley graph [closed]

Consider the Cayley graph of a finite non abelian group, generated by a generating set, say $\{s,t\}$. Let me explain my question by thinking of a small example. Let the Cayley graph have 5 vertices, $...
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2answers
48 views

Classification of groups of order $p^2q$, when $q|p+1$

When considering non-abelian groups of order $p^2q$, where $p>q$ and $p,q>3$, two cases can be identified, when $q|p^2-1$. Case 1: $q |p+1, q \nmid p-1$ Case 2: $q |p-1, q \nmid p+1$ Can someone ...
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1answer
30 views

A simple equality related to an irreducible representation of a finite group

Let $G$ be a finite group of cardinal $|G|$. Let $u \colon G \to M_n$ be an irreducible unitary representation of $G$. How to show that for any matrix $A \in M_n$ we have $$ \frac{1}{|G|} \sum_{g \in ...
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1answer
45 views

Complete reducibility of group representation [closed]

Let $\rho$ be a finite-dimensional representation of a finite group $G$ over the field $\mathbb{F}$. Suppose that $N$ is a normal subgroup in $G$ and the characteristic of $\mathbb{F}$ does not divide ...
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1answer
44 views

Given $\sigma$ a coprime automorphism of $G$ a finite group, must the centre of $G^\sigma$ be contained in the centre of $G$?

Let $G$ be a finite group, $\sigma$ an automorphism of order coprime to $|G|$. Let $K$ denote the subgroup of $\sigma$ fixed points of $G$. Then do we necessarily have that every element of $G$ ...
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1answer
93 views

$D_{2n}$ is isomorphic to a subgroup of $S_n$ (for $n>2$).

All the proofs I've come across of the fact in the title call into play the action of the group on the vertices of the regular $n$-gon, i.e. they rely on the geometrical definition of $D_{2n}$ (in ...
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1answer
41 views

All idempotents are central then $KG$ has no nilpotent

Given $G$ a finite group and $K$ a field of characteristic zero such that all idempotents in $KG$ are central, is it true that $KG$ has no nilpotent element or equivalently $KG$ has only division ring ...
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2answers
64 views

“The p-power bounding” right multiplication in a finite group G under some special conditions

[Roughly speaking, the following question considers a special setting in which we want to prove a property in the form of $ord(g \sigma)\ |\ p^k$.] The Problem in Detail: Let $G$ be a finite group, ...
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1answer
36 views

Sufficient condition for $\mathrm{ord}(a)$ divides exponent of finite abelian group?

In Niederreiter and Winterhof, Applied Number Theory, Proposition 1.3.24 (p. 18) is: If $G$ is a finite abelian group of exponent $E$, then $\mathrm{ord}(a)$ divides $E$ for all $a \in G$. They're ...
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1answer
44 views

Known Plaintext-like attack on Multiplicative Group-based cipher

I have been presented with a problem where I need to find a key $k$, which is the element of a Multiplicative Group $\Bbb Z_p^*$, where p is known. I further have three pairs of the form $(a_i, k*g^{...
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32 views

Are all group actions on the simplex permutations?

On the standard $n$-simplex $\Delta[n]$ in $\mathbb{R}^{n+1}$ there is the usual action of the symmetric group $S_n$, by permuting the vertices. Any element $g\in S_n$ then acts on $\Delta[n]$ ...
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1answer
43 views

Does every set of generators of a finite group contain a minimal set of generators? [closed]

Suppose that $G$ is a finite group that can be generated from $n$ elements $g_1,...,g_n\in G$. Now, if $S\subseteq G$ is another set of generators of $G$, can I always find a subset $S'\subseteq S$ of ...
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38 views

A exercise in basic algebra Cohn

This question appears in Basic Algebra written by Cohn and really stuck on me: Let $G$ be a group and $K, L, N$ be subgroups such that $N$ is a normal subgroup of $K$, $K$ is a normal subgroup of $G$,...
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1answer
53 views

Subgroups of product of two finite groups with coprime order

I want to explain the following proposition (if true). Let $G_1$ and $G_2$ be finite groups with coprime orders and let $H$ be a subgroup of $G_1 \times G_2$. Then there exists subgroups $H_1$ and $...
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1answer
32 views

What is the number of degree $1$ representations of a finite group?

Let $G$ be a finite group and $G'$ be the derived subgroup of $G.$ Then I know that every degree $1$ representation of $G$ factors through $G/G'.$ How does it imply that the number of degree $1$ ...
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49 views

Definition of $PSL_2(\mathbb{F}_7)$

The following is an excerpt of Serre's book on finite groups. I am very confused by his definition of the group $PSL_2(\mathbb{F}_7)$. Why does he take $ad-bc$ a non-zero square? The definition I know ...
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1answer
80 views

Dihedral group $D_4$ (symmetries of a square) is isomorphic to a subgroup of $S_4$ (permutation group) [duplicate]

I have this problem that I have been stuck on for a while. I know that I can write the elements of $D_4$ as products of the identity, rotation by $90$ degrees and a vertical reflection and I have ...
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0answers
28 views

Do we have to specify actions by many elements when defining some semidirect products

When studying about the semidirect product, $G=(\mathbb{Z}_p \times \mathbb{Z}_p)\rtimes_{\phi} \mathbb{Z}_q$, I understood that, for some semidirect products by considering a minimal generating set $...
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3answers
38 views

$p$-subgroup of Sylow and Normalizer

Let $G$ be a finite group, $p$ a prime that divides $|G|$, and $P$ a $p$-subgroup of Sylow of $G$. Show that $P$ is the unique $p$-subgroup of Sylow that is in $N_G(P)$. I tried this by assuming there ...
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3answers
92 views

The number of groups of order 32

There are 51 groups of order ($32=2^5$). My question is how this number was computed. Graham Higman and Charles Sims gives an estimate for the number of $p$-goups (i.e. groups of order $p^n$ where $p$-...

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