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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Cayley's Theorem - Discourse on Fraleigh's proof or Alternative Easier proof [duplicate]

I have hard time understanding Fraleigh's proof. Can someone either explain Fraleigh's or provide an alternative, perhaps easier proof? Thanks so much.
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5answers
433 views

Let $G$ a group of order $6$. Prove that $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$.

Let $G$ a group of order $6$. Prove that: i) $G$ contains 1 or 3 elements of order 2. ii) $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$. I haven´t covered Sylow groups and normal groups. ...
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5answers
273 views

What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$

What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$ and how to find it? I came across the above problem and do not know how to get it? Can someone point me ...
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3answers
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Is $PGL_2(q)$ isomorphic to $SL_2(q)$

Let $F_q$ denote the field of order $q$. Define: $GL_2(q)$ to be the group of invertible $2$ by $2$ matrices over $F_q$. $SL_2(q)$ to be its subgroup consisiting of invertible $2$ by $2$ matrices ...
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Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.

Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic. What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve ...
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1answer
237 views

Right translation - left coset - orbits

We can remark that the left coset $gH$ of $g \in G$ relative to a subgroup $H$ of $G$ is the orbit of $g$ under the action of $H \subset G$ acting by right translation. What is that right ...
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238 views

Group theory - left/right $H$-cosets and quotient sets $G/H$ and $G \setminus H$.

Let $G$ be a group and $H$ be a subgroup of $G$. The left $H$-cosets are the sets $gH, g \in G$. The set of left $H$-cosets is the quotient set $G/H$. The right $H$-cosets are the sets $Hg, g\in G$. ...
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1answer
676 views

Intuition behind Maschke's theorem

I'm an undergraduate learning about group representations and Young tableaux, and have came across Maschke's theorem stating; If $G$ is a finite group and $F$ is a field who's characteristic does ...
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0answers
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characteristic-$p$-type groups, and the Borel-Tits theorem for $PSL(V)$

If $G$ is a finite group and $F^*(G)$ is the generalized Fitting subgroup we say that $G$ has characteristic $p$ ($p$ is a prime that divides $|G|$) if $$F^*(G)=O_p(G)$$ Moreover $G$ is a ...
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1answer
603 views

Let $\alpha$ be a $2$-cycle and $\beta$ be a $t$-cycle in $S_n$. Prove that $\alpha\beta\alpha$ is a $t$-cycle

$\bf Claim:$ Let $\alpha$ be a $2$-cycle and $\beta$ be a $t$-cycle in $S_n$. Prove that $\alpha\beta\alpha$ is a $t$-cycle. If $\alpha$ and $\beta$ are disjoint, they commute and thus the product ...
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5answers
199 views

Find all the subgroups of $\mathbb{Z}_7$ and $\mathbb{Z}_9^\times$

Can you please help me in this question: Find all the subgroups of $\mathbb{Z}_7$ and $\mathbb{Z}_9^\times$. Thanks a lot
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3answers
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A question on finite $p$-groups. [duplicate]

Is true that if $G$ is a $p$-group finite, say, $\mid G \mid = p^d$, then $G$ is $d$-generated?
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Normal Subgroups, index, divisible orders

Let $H$ be a normal subgroup of $G$ with index $k$ . Show that if $a \in G$ and $o(a)=n$, then the order of $aH$ in $G/H$ divides both $n$ and $k$ .
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Order of a permutation group

I'm playing with a permutation group, with generators $(1,2,3)(6,5,4)$ and $(2,5,7)(8,6,3)$, in cycle notation. After careful counting, I believe its order is 24. But I have no real method, except ...
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1answer
197 views

Definition of induced representation by tensor product

Suppose there is a finite group $G$ with a subgroup $H$ an some field $K$. If one has a representation of the subgroup, one can construct the induced representation $\rho:=Ind_H^G$ according to ...
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1answer
168 views

Systems of coset representatives

I have the following question. Let $H\leq U\leq G$ be (not necessary finite) groups. Let $S$ is a System of Right coset represantatives of $U$ in $G$, i.e. $\bigcup_{s\in R} Us=G$ with $Us\cap ...
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1answer
28 views

$m'$-group being cyclic?

Given a group $G=\mathbb{Z}_m\rtimes\mathbb{Z}_n$ with $m,n$ coprime. Should every subgroup of $G$ that has order coprime to $m$ be cyclic?
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1answer
339 views

Sylow $p$-Subgroups of Classical Groups over $\mathbb{F}_p$

Let $p$ be a prime, and let $G$ be any of the finite classical groups $SL_n(\mathbb{F}_p)$, $O_n(\mathbb{F}_p)$, or $SP_n(\mathbb{F}_p)$. Let $P$ be a Sylow $p$-subgroup of $G$. What is $P$ as a ...
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Groups not arising from certain centralizers

There's a lot of fuss in certain subfields of algebraic topology about giving fancy interpretations to the rings coming from the cohomology of groups, where "cohomology" is allowed to be taken to be ...
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Some groups of order $40$

Is there some table on the web giving information about particular small groups, that would go up to order $40$ and that would give enough information so that one could be sure whether groups matching ...
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2answers
406 views

Order of kernel of a homomorphism.

Let $C_n$ denotes the cyclic group of order $n$ and let $\phi:C_{52}\rightarrow C_{52}$ be the homomorphism $\phi(x)=x^7$. What is the order of kernel of $\phi$? I know that $ker\phi=\left\{x/ ...
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1answer
220 views

Exercise 2.15 M.Isaacs' Character theory of finite groups

I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book. I would need some help with this one: (2.15): Let $\chi\in \operatorname{Irr}(G)$ be ...
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1answer
285 views

Exercise 2.8 M.Isaacs' Character theory of finite groups

I'm a starter at character theory. I'm trying to do this exercise: (2.8) Let $\chi$ be a faithful character of a group $G$. Show that $H\subseteq G $ is abelian if and only if every irreducible ...
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1answer
252 views

Character theory exercises [closed]

I'm doing the exercises from chapter 2 of M.Isaacs' Character theory of finite groups, and I'm having problems with some of them. In particular, I would need help with these ones. Thank you very much ...
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2answers
344 views

Let $G$ be a finite group and $P$ be a Sylow $p$-subgroups of $G$. Let $H$ be a subgroup of $G$ such that $N_G(P)\subset H$. Prove that $N_G(H)=H$

1.Let $G$ be a finite group and $P$ be a Sylow $p$-subgroups of $G$. Let $H$ be a subgroup of $G$ such that $N_G(P)\subset H$. Prove that $N_G(H)=H$ 2.Let $G$ be a group of order $143$. Show that ...
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1answer
187 views

Question about the fundamental group of simplicial complex and the universal cover.

Can any one give me any idea about how to solve this problem? Suppose we have a simplicial complex G which is finite connected. (1)The fundamental group of G is finite; (2)The universal cover of G is ...
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3answers
174 views

How do I find $\left|\langle a,b\mid a^2=b^3=e\rangle\right|$?

Suppose $G$ is a group satisfying $G=\langle a,b\mid a^2=b^3=e\rangle$. Find $|G|$.
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Prove that there exist no element of order 18 in $S_9$

Prove that there exist no element of order 18 in $S_9$. How do I prove this ? I think the idea is that elements of the form: $(123456)(789)$ have order 6 as $\text{lcm}(6,3)=6$. Elements of the form ...
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1answer
75 views

Sylow $p$-subgroups of $SO_3(\mathbb{F}_p)$

Let $G=SO_3(\mathbb{F}_p)$. We have $|G|=p(p^2-1)$. Let $n_p$ be the number of Sylow $p$-subgroups of $G$. Is it true that $n_p=p+1$? We have the two conditions $n_p\equiv 1\mod p$ $n_p\mid ...
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3answers
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at least one element fixed by all the group

$G$ is a p-group and $S$ is a set that $G$ acts on. p does not divide $|S|$. Why is there at least one element $a\in S$ such that $|O(a)|=1$, or in other words, $G_a=G$? I tried to ask this question ...
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Sylow $p$-subgroups of Finite Matrix Groups

Let $G$ be a subgroup of $GL_n(\mathbb{F}_p)$, with $n\le p$, and let $P$ be a Sylow $p$-subgroup of $G$. Do all non-trivial elements of $P$ have order $p$? I believe the answer is yes, because I ...
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Let $\sigma,\tau \in S_n$. Prove that $\sigma \tau$ and $\tau \sigma $ have the same cycle type.

Let $\sigma,\tau \in S_n$. Prove that $\sigma \tau$ and $\tau \sigma $ have the same cycle type. I was thinking that you could rewrite $\sigma=g_1\cdots g_k$ with $g_i$ disjoint cycles and ...
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2answers
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A nontrivial p-group has nontrivial center

I know this is a very common corollary of the class equation. And I know how to do it by using class equation. But can you do it bu using group action, maybe find a nice set for G to act on?
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1answer
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Solvability of a group with order $p^n$

If $G$ is a group whose order is $p^n$($p$ is prime), then $G$ is solvable. How am I going to show this? Any help is appreciated. Thank you.
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1answer
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Group Isomorphism

I'm trying to show the last part to the following question: Every element in a group $G$ has order $2$, prove $G$ is abelian. Show that if $H$ is a subgroup of G and $g \in G\backslash H$ then $K ...
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1answer
86 views

A question about homomorphisms and the existence of a subgroup.

On page 117 of (Corollary 4.6.12), http://www.albany.edu/~mark/algebra.pdf I'm not sure how/why the proof says that there exists "a unique subgroup of order 5." How did they come up with that ...
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0answers
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Group extension analysis

Let $\mathbb{Z}_p\lhd H\leq\text{AGL}(1,p),\mathbb{Z}_q\lhd K\leq\text{AGL}(1,q)$ with $p,q$ prime. Let $G=H.K$. Can one show that $G$ contains a normal subgroup of order $pq$? Note: Here $G=H.K$ ...
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3answers
189 views

Formula for Product of Subgroups of $\mathbb Z$, Problem

What is the product of $\mathbb{Z}_2$ and $\mathbb{Z}_5$ as subgroups of $\mathbb{Z}_6$? Since $\mathbb{Z}_n$ is abelian, any subgroup should be normal. From my understanding of the subgroup product, ...
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1answer
63 views

Maps of representations

Let $G$ be a group and $V_j$ , $j =1,2$ be irreducible representations of $G$. Show that any map $\phi: V_1 \rightarrow V_2$ of representations is either an isomorphism or zero. The Hint I got was: ...
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1answer
167 views

How to prove one-one and onto?

Let $G$ be a group of odd order. Show that the function $\phi:G \to G $ given by $\phi(g)=g^2$ is one-one and onto. To prove one-one, I did $\phi(g_1)=\phi(g_2)$ implies $g_1^2=g_2^2$. Somehow I got ...
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2answers
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Do group rings appear outside of representation theory?

I am particularly concerned with finite groups. I have seen group rings used in the fundamentals of representation theory as the dual notion to representations. I haven't ever seen them anywhere ...
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1answer
527 views

Literature on group theory of Rubik's Cube

While searching for literature on the group theory of Rubik's Cube, I mostly find introductions to group theory motivated by applications to Rubik's cube. I.e. the focus lies on elementary group ...
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1answer
177 views

Equivalence relation on a group

I have the following question: Let $G$ be a finite group. We define a relation $\sim$ on $G \backslash \left\{e\right\} = \left\{ g \in G : g \neq e \right\}$ by $g \sim h$ if and only if there ...
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3answers
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Is there a quick trick to write permutations of $S_n$ as products of transpositions?

If I want to write $(123)$ as product of transpositions, I get $(13)(12)$. For $(132)$ I get $(12)(13)$. For $(1234)$, I get $(14)(13)(12)$. Seems like I can write $(abcd)$ as $(ad)(ac)(ab)$. Is this ...
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1answer
93 views

Natural way to define a free action of a finite abelian group

Let $G$ be a finite abelian group. Then $G \simeq \mathbb{Z}_{u_1} \oplus \cdots \oplus \mathbb{Z}_{u_m}$, where $u_{i}$ is a power of some prime number. Without loss of generality I will consider $G ...
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3answers
63 views

Given order of x find order of

Given the order of $x=36$ in a group, how do I compute the order of $x^{-8},x^{27} $. Also, a similar question, for $x, y \in G$, if order of $x=2$ and order of $y=3$, what can we say about order of ...
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1answer
309 views

A doubt in the proof of Frucht's theorem

I am trying to understand the proof of Frucht's theorem which is: Every finite group is isomorphic to the automorphism group of some simple graph. The proof (which I am reading from this book) ...
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1answer
89 views

Construct a complete system of representatives for the left cosets of $H_2$ in $G$.

Let $G$ be a finite group. Let $H_2\subseteq H_1$ be subgroups. Let $R$ be a complete system of representatives for the left cosets of $H_1$ in $G$. Let $S$ be a complete system of representatives ...
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1answer
122 views

Union of Cosets as a Subgroup

Let $H$ be a proper subgroup of finite group $G$ such that $p.|H|<|G|$. Let $x\in G\setminus H$ be an element of order $p$. Under what conditions, the set $H\langle x\rangle = H\cup Hx\cup \cdots ...
32
votes
2answers
485 views

Are there/Why aren't there any simple groups with orders like this?

The orders of the simple groups (ignoring the matrix groups for which the problem is solved) all seem to be a lot like this: ...