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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1answer
87 views

Central Series - Normal subgroups

$H$, $N$ are subnormal subgroups in the finite group $G$ and $G = H*N$. Show: $(H*N)^{\infty} = H^{\infty}*N^{\infty}$. (And $G^{\infty} := \bigcap\limits_{i\geq 0}G^{i}$, and $G^{i+1} = [G, G^{i}]$ ...
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1answer
36 views

How can I show that $ASL_n(F)$ is acting 2-transitively?

One of my friends asked me to ask this question here. This is a question from his last exam: Let $$ASL_n(F)=\{T_{A,v}:V_n(F)\to V_n(F)\mid\exists A\in SL_n(F), \exists v\in V_n(F), ...
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0answers
77 views

Smooth Action of a Finite Group

Suppose $H$ is a finite group acting smoothly on a smooth connected manifold $M$. The action is trivially proper, as $H$ is discrete. If the action of $H$ were also known to be free, i.e. $h\cdot ...
2
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1answer
188 views

Proof that cofactor expansion has unique value

Edit: I'm genuinely not sure why this has gotten little activity. If somebody knows, please tell me, so I can rework it. As a note: I am a purist, and really want to see a proof of this, but I've ...
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1answer
39 views

$G = \mathbb{Z}_{n_1}\times\cdots\times\mathbb{Z}_{n_k}$ contains a element of order $m$ iff $m\mid n_1$.

Let $G$ be a finite Abelian group: $$G=\mathbb{Z}_{n_1}\times\cdots\times\mathbb{Z}_{n_k}$$ and $n_k \mid n_{k-1} \mid \cdots \mid n_1$. Show that $G$ contains an element of order $m$ iff $m\mid ...
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1answer
120 views

Why is $(1,2,…,p)$ in the center of a Sylow $p$-subgroup of $S_n$?

Assuming $p$ divides $n$, let $P$ be a Sylow $p$-subgroup of $S_n$ and let $z=(1,2,...,p)$. Why is $z$ in the center of $P$? Thanks!
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49 views

Subgroups of $G^n$

Let $G$ be a non-Abelian group and let $n$ be a (large) positive integer (eventually going to infinity). For simplicity, let's take $G=\mathbb{D}_6$ (Dihedral group of order $6$). Is there a way to ...
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1answer
60 views

Composition series of $\mathbb F_{p^k}$

I am trying to find the composition series of the group $\mathbb F_{p^k}$, where $p$ is prime , and $k\ge 1$. From Jordan Hölder equation it has length $k$ , i am quite confused , It looks quite ...
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1answer
195 views

Multiplicity of a completely reducible representation in another irreducible representation.

I have got the next question that I am pondering the answer to. Let $\tau$ be a completely reducible representation of finite dimension of a group $G$, and let $\pi$ be another irreducible ...
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2answers
95 views

$G=\mathbb{Z}_{60}\times\mathbb{Z}_{45}\times\mathbb{Z}_{12}\times\mathbb{Z}_{36}$

Let $$G=\mathbb{Z}_{60}\times\mathbb{Z}_{45}\times\mathbb{Z}_{12}\times\mathbb{Z}_{36}$$ Determine number of elements have order of $2$, and number of subgroups have order of $2$.
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1answer
51 views

Prime divisor in a finite group

If a prime number $p$ divide the number of elements of order $k$ (for some $k\neq p$) in a finite group $G$, then whether we can say that $p$ divide order of $G$?
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2answers
112 views

Composite group homomorphism between alternating groups

Let $N$ a non-trivial normal subgroup of $A_n$ and $H = N \cap A_{n-1}$. I would like to show that $A_{n-1} \hookrightarrow A_n \to A_n/N$ is surjective, where $A_n \to A_n/N$ is the canonical ...
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2answers
356 views

Klein 4 group is the only proper normal subgroup in $A_4$

How do you show that the only nontrivial normal subgroup of $A_4$, which is also not the whole group is the Klein 4 group, denoted by $V$ (or isomorphic to the Klein 4 group)? I've shown before that ...
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1answer
109 views

Normal subgroups of the Alternating group $A_n$

Is $A_{n-1}$ a normal subgroup of $A_n$? If so, why?
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3answers
429 views

Presentation of Rubik's Cube group

The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
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4answers
128 views

Center and Fitting subgroups

I have to prove that, if G is a finite soluble and nonabelian group, its center is a proper subgroup of the Fitting subgroup of G. In other words, that $Z(G)<F(G)$ Any ideas? Thanks a lot.
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2answers
250 views

Transitive action of normal subgroup of the alternating group

everyone! Would anyone be willing to give me any sort of help with the following question? Let $n\ge 4$ and $A_n$ the alternating group. Let $N$ a non-trivial normal subgroup of $A_n$. Prove that the ...
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2answers
63 views

Cardinality of a subset acted upon by the Alternating Group, $A_n$

Suppose $X \subset \{1,2,3,\ldots,n\}$. Show that the cardinality of $X$ is $0$, $1$ or $n$, if $\forall$$b \in A_n$, $X \cap bX = \emptyset$ or $X = bX$. It's pretty clear to me how the cardinality ...
1
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1answer
249 views

Notations in Group theory

I will start by apologizing as many will not like this question. I am reading the paper COHOMOLOGY THEORY OF GROUPS WITH A SINGLE DEFINING RELATION and having focused on typology throughout my ...
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2answers
211 views

$G/N$ is nilpotent $\Leftrightarrow$ $N$ contains the intersection of all subgroups in the lower central series.

I want to solve the following exercise: Let $G$ be a group. $G^{0}\geq G^{1} \geq G^{2}\dots$ is a lower central series, s.t. $G^{0} = G$ and $G^{i+1} = [G,G^{i}]$. Let $N$ be a normal subgroup of ...
1
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1answer
101 views

The $h-$th power of every element in a finite group of order $h$ is the identity element of the group

I work in population dynamics. I want to show somewhere in my work that the $h-$th power of every element in a finite group of order $h$ is the identity element of the group. I guess this is ...
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1answer
69 views

Need clear up on subgroup and quotient group

Consider $26 + <12>$ in $\mathbb{Z}_{60} / <12>$, I need to find the order of the element in this quotient group. So I basically wrote out what the group might look like ...
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1answer
202 views

Examples of finite groups that are generated by elements $x$ satisfying $x^n = 1$

Let $G$ be a finite group and $n > 1$ a divisor of $|G|$. Let $P_n(G)$ be the subgroup of $G$ which is generated by all elements which satisfy $x^n = 1$. Since $n > 1$, by Cauchy's theorem we ...
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0answers
76 views

Question on restriction of irreducible group representations to normal subgroups

I am confused about the answers to the following question: Restriction to a normal subgroup with the original question copied here: Let $A$ be a normal subgroup of a finite group $G$ and $V$ an ...
6
votes
1answer
327 views

Find all permutations that commute with $\omega$=(1 9 7 10 12 2 5)(4 11)(3 6 8) in $S_{12}$

I'm asked in this exercise to find all permutations that commute with $\omega$=(1 9 7 10 12 2 5)(4 11)(3 6 8) in $S_{12}$. What I have so far: We could write $x$(1 9 7 10 12 2 5)(3 6 8)=(1 9 7 10 12 ...
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1answer
71 views

A noneempty subset $H$ of a finite goup $G$ is a subgroup if and only if $H$ is closed

I need to prove proposition 1, and this is what I have. Definition 1: If there exists a nonzero integer such that $a^m=e$, then the order of the element $'a'$ is defined to be the least positive ...
3
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1answer
121 views

A faster way to show that a subgroup is normal

I'm working with $\mathbb S_4$, and I have a subgroup of $\mathbb S_4$ called $G$. $G$ is generated by $a=(12)(34)$ and $b=(123)$, which I've actually found to be $A_4$ by multiplying elements by $a$ ...
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0answers
41 views

Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

I'm new to algebaric groups, so please accept my apology in advance if I post crazy questions. Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ ...
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2answers
55 views

Checking that $\rho$ is a representation

Let $z$ be a generator of the cyclic group $\mathbb{Z}_3 = \{ 1,z,z^2 \}$. Prove that a representation $\rho$ of $\mathbb{Z}_3$ in the $2$-dimensional complex vector space $\mathbb{C}^2$ can be ...
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1answer
72 views

A group with three maximal abelian subgroups

I am looking for a group that has exactly three maximal abelian subgroups. I thought about the quaternion group. $G=Q_8 = \langle x,y \mid x^4=1, x^2=y^2, yxy^{-1}=x^{-1}\rangle$. $Z(G) = ...
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0answers
101 views

Small question about nonabelian simple group.

My doctor gave me the following theorem: Theorem: Suppose that $G$ is a finite non-abelian simple group. Then there exists an odd prime $p \in \pi (G)$ such that $G$ has no $\{2,p\}$-Hall subgroup. ...
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3answers
146 views

$S_n$ acting transitively on $\{1, 2, \dots, n\}$

I am reading Dummit and Foote, and in Section 4.1: Group Actions and Permutation Representations they give the following example of a group action: The symmetric group $G = S_n$ acts transitively ...
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2answers
703 views

Given 3 distinct primes {$p,q,r$}, then $|G|=pqr \implies G$ not simple

Here's a question I've been asked; Given distinct primes $p,q,r$, show that any group $G$ of order $pqr$ is not simple. So far, my idea has been to individually check each possible proper subgroup, ...
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1answer
162 views

Chinese remainder theorem and isomorphism

Suppose that there are different primes $p_1$ and $p_2$ and the group $C_{p_1}$. I would like to decompose the following tensorproduct and thought of using the chinese remainder theorem to get ...
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3answers
168 views

Prove that $S_n$ is doubly transitive on $\{1, 2,…, n\}$ for all $n \geqslant 2$.

Prove that $S_n$ is doubly transitive on $\{1, 2,\ldots, n\}$ for all $n \geqslant 2$. I understand that transitive implies only one orbit, but...
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2answers
277 views

Isomorphisms between finite abelian groups

I am working on the following problem: Are there any isomorphisms between the following finite abelian groups?: $$\mathbb{Z}_{1225}\times\mathbb{Z}_{315},$$ ...
2
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4answers
288 views

Dihedral group $D_{8}$ as a semidirect product $V\rtimes C_2$?

How do I show that the dihedral group $D_{8}$ (order $8$) is a semidirect product $V\rtimes \left\langle \alpha \right\rangle $, where $V$ is Klein group and $% \alpha $ is an automorphism of order ...
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0answers
128 views

Properties Socle

$\newcommand{\Soc}{\operatorname{Soc}}$ I am trying to solve why this holds: $N$ normal subgroup of $G$ (finite). $\Soc(\Soc(N)/(N \cap \Soc(G))) = \Soc(N)/(N \cap \Soc(G))$. As I could not solve ...
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0answers
65 views

normal subgroup + Socle

I am looking for examples of socle and normal-subgroup relations. If $G=S_{4}$, the normal subgroups are $A_{4}$ and $V_{4}$, thus the socle of $G$ is the klein-four group, and for $A_{4} \cap Soc(G) ...
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2answers
33 views

Constructing a group with subsets

Given the set $S=\{0,1,2\}$, it has been asked to prove that it is not a group under the operation $\max(x,y)$. It can be done. Then they ask to identify $3$ subsets of $S$ which are 'groups' under ...
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1answer
80 views

$G/G' \cong I/I^2$ where $I$ is the augmentation ideal [duplicate]

Possible Duplicate: Isomorphism between $I_G/I_G^2$ and $G/G'$ Let $G$ be a finite group. Let $I\unlhd\mathbb{Z}[G]$ be the augmentation ideal of the integral group ring $\mathbb{Z}[G]$. ...
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3answers
89 views

A step in the proof of Cauchy's theorem for groups

Let $G$ a group, $p$ a prime and $X=G^p$. Let $\sigma\in S_X$ act as follows: $\sigma(x_1,...,x_p) = (x_2,...,x_p,x_1)$. Let $Y$ the subset of elements in $X$ such that $x_1x_2...x_p=1$ and let ...
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1answer
119 views

Injective Homomorphism on $S_n$

I have the following question on a homomorphism between symmetric groups and it has been really a pain. I know I am supposed to use induction but, I seem to miss something essential so if anybody ...
2
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1answer
68 views

The indecomposable projective $\mathbb{F}_pG$-module with $UJ/J\cong \mathbb{F}_p$

Let: $G$ be a finite group; $p$ be prime; $J$ be the Jacobson radical of $\mathbb{F}_pG$. A paper I'm trying to read mentions the following object: The indecomposable projective ...
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126 views

Regular subgroups of affine general linear groups

Let $AGL(2d,2)$ be the affine general linear group acting natrually on a $2d$-dimensional vector space over $GF(2)$. Is there a regular subgroup of $AGL(2d,2)$ isomorphic to $Z_{2^d}:Z_{2^d}$ for ...
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2answers
4k views

More than 99% of groups of order less than 2000 are of order 1024?

In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024. Is this for real? How can one deduce this result? ...
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0answers
87 views

Product of subgroups and generating sets

Prove or disprove the following: $(1)$ Let $H$,$K$ be subgroups of a finite group $G$. If $|HK|=|\langle H,K \rangle|$, then $HK=\langle H,K \rangle$, that is, $HK$ is a subgroup of $G$. $(2)$ Let ...
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0answers
104 views

Hall subgroup of a finite group?

How did the author get that $L=(L \cap H)(L\cap K)$ in Lemma $5$ below. Remark: All the groups here are finite. $H$ permutes (commutes) with $K$ means $HK=KH$ where $H$ and $K$ are subgroups of some ...
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0answers
62 views

Soluble subgroups of finite classical simple groups

What do we know about the soluble subgroups of finite classical simple groups (and maybe their automorphism groups)? For example, are their maximal soluble subgroups explicitly known?
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2answers
206 views

Does a Frobenius group with a $p$-group complement necessarily have a normal Sylow $2$-subgroup?

Let $G=KH$ be a Frobenius group of even order with Frobenius kernel $K$ and Frobenius complement $H$ such that $\pi(H)=\{p\}$, where $p$ is prime. Why is a Sylow $2$-subgroup of $G$ normal in $G$?