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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Ordered pairs of permutations in symmetric group

How many ordered pairs $\left(\alpha_1,\alpha_2\right)$ of permutations in symmetric group $S_n$ that commute: $$\alpha _1 \circ \alpha _2 = \alpha _2 \circ \alpha _1\,,$$ where $\alpha _1, \alpha _2 ...
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1answer
20 views

Order of multypling 2 sub-groups who's orders are coprime

Im given as an exercize to prove that an order of 2 sub-groups A,B who's orders are coprime, is: $$|A| \cdot |B|$$ What I know that generally: $$|AB|=\frac{\left|A\right|\cdot ...
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0answers
16 views

characterization of all finite subgroups of mobius transformation

does there exist any characterization of all finite subgroups of the group $G$, where $G$ is the group of automorphisms of the open unit disk ?
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1answer
12 views

conditions on multypling sub-groups (so it would be a new sub-group)

Ok, so we know that if G is abelian and A,B are here sub-groups, then AB, defined by: $AB\:=\:\left\{ab\::\:a\in A,\:b\in B\right\}$ is a new sub-group. Now, I'm given another condition and I need ...
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2answers
82 views

$G$ finite, the number of distinct conjugates of $x$ is the index of the normalizer $N_x$ of $\{x\}$ in $G$

In order to prove this, I did the following: first, I showed that conjugacy forms an equivalence relation, then I can find its conjugacy classes. I understand how to form a conjugacy class, given a ...
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1answer
20 views

$ M_{G} $ is the core of $ M $ in $ G $. Let $ M_{G} = 1 $. Why $ M $ is a complement for $ N $ in $ G $?

Let $ M $ maximal subgroup of solvable group $ G $, and assume that $ G = MC $ for some cyclic subgroup $ C $. Let $ N $ be a minimal normal subgroup of $ G $, then $ N $ is an elementary abelian $ p ...
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1answer
17 views

$ N $ is an elementary abelian $ p $-subgroup. Is true $ N $ is cyclic group? $

Let $ G $ is solvable group and $ M $ be a maximal subgroup of $ G $. Let $ N $ be a minimal normal subgroup of $ G $, then $ N $ is an elementary abelian $ p $-subgroup. Is true $ N $ is cyclic ...
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3answers
131 views

$G$ has order $p^a$, then the center of $G$ counts more than the identity

This question makes no sense to me. I don't know what it means by "counts more than the identity". Then, the exercise gives me a hint: "break G into equivalence classes of conjugacy elements and see ...
4
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1answer
48 views

Prove $[N(H):H]\equiv [G:H](\mod p)$

Let $G$ be a finite group and $H$ be a subgroup of $G$. Let $|H|=p^n$ for some $p$ prime, $n\geq1$. Show that $[N(H):H]\equiv [G:H](\mod p)$. I observed that I will need to show that $p$ divides ...
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1answer
35 views

Picking out a subset of elements from a finite product of cyclic groups

Let $C_n$ be the cyclic group of order $n$, and let $G = \prod_{i=1}^n C_n = \underbrace{C_n \times C_n \times \ldots \times C_n}_{n \text{ times}}$. For $g = (g_1,g_2,\ldots, g_n) \in G$, call $g$ ...
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0answers
42 views

Is it true $ N(P \cap M) = NP \cap NM = N(NC) \cap G = P \cap G = P $?

Let $ M $ be a maximal subgroup of a solvable group $ G $, and assume that $ G = MC $, for some cyclic Subgroup $ C $. Let $ N $ be a minimal normal subgroup of $ G $. Then $ N $ is an elementary ...
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1answer
58 views

Assume that $ G = MC $, for some cyclic subgroup $ C $. Is $ M \cap C $ a normal subgroup of $ G $?

Let $ G $ is a solvable finite group and $ M $ be a maximal subgroup of $ G $, and assume that $ G = MC $, for some cyclic subgroup $ C $. If $ M_{G} = 1 $ that $ M_{G} $ is core of $ M $ in $ G $, ...
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0answers
34 views

Let $ M $ be a maximal subgroup of a solvable group $ G $, and assume that $ G=MC $, for some cyclic subgroup $ C $.

Let $ M $ be a maximal subgroup of a solvable group $ G $, and assume that $ G=MC $, for some cyclic subgroup $ C $. Then $ \vert G : M \vert $ is a prime or $ 4 $. Also if $ \vert G : M \vert = 4 $ , ...
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3answers
88 views

Show that $G$ is cyclic

Here is a problem from Herstein. Let G be a finite abelian group so that the equation $x^n=e$ has at most $n$ solutions in $G$ for every positive integer $n$. Show that $G$ is cyclic. I will ...
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2answers
62 views

The difference between Z(G) and C(a) in an example

I found that I didnt understood the defenitions. I have this exercize: to prove that $a\in Z(G)$ $<==>$ $C(a) = G$ Is there here something to prove? Isnt it directly of their defenitions ? I ...
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1answer
46 views

Is this subset of GL(C) a sub-group?

$$G\:=\:GL_2\left(R\right)$$ $$\:N=\left\{A\in G\:;\:A\cdot A^T=I\right\}$$ Is N a subgroup of G? So the main work here is to prove closure for the opposite. We want $A^{-1}\in ...
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1answer
54 views

Does there exist a group of order $ p^\alpha q^\beta$ that is simple?

I am working on an problem that calls for the existance of simple group of order $p^\alpha q^\beta$ with $\alpha , \beta \geq 1$. I was wondering if such a group existed. Edit: The problem I was ...
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1answer
25 views

G finite group with the following characteristics

G is closed finite group with an associative operator with the following parametres/characterstics: For each $x,y\in G$ if $ax = ay$ then $x = y$ for each $a\in G$ For each $w,z\in G$ if $wa = za$ ...
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3answers
62 views

G is finite group. Need to proof that exists natural k that $g^k = e$ [duplicate]

How do I prove that in a finite group G, for each element in G there is natural power (say $k$) which depends on g,such that $g^k=e$ ? I need to show the existence and the dependence on which $g$ I ...
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1answer
40 views

Unit cancellation in group rings

Suppose I have a finite group $G$, a non-trivial proper subgroup $H$, a field $k$ (restricting to $k=\mathbb C$ would be fine), and non-zero elements $a,u$ in the group algebra $kG$ satisfying the ...
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1answer
45 views

Normal group that contains its centralizer

I am studying for my Algebra qual and I came across this question: Let $G$ be a finite group with a normal subgroup $N$ such that $C_G (N) \leq N$. Show that $$ |G|\leq |N|!. $$ Here $C_G (N)$ is ...
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2answers
32 views

Let $G$ be a group of all $2 \times 2$ matrices over $Z_p$ with determinant $1$ under matrix multiplication. To find the order of $G$.

I am solving some previous year's question paper of our college and found the following problem: Let $p$ be a prime number. Let $G$ be a group of all $2 \times 2$ matrices over $Z_p$ with determinant ...
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2answers
40 views

A generator of intersection of two sub-groups.

How do I find it? For example, an easy one: $G\:=\:\left(\mathbb{Z},+\right)$ H and K are sub-groups: $n\mathbb{Z}, m\mathbb{Z}$ for different $n$ and $m$. And we know that $n$ and $m$ are ...
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1answer
16 views

Can we embed every finite group in some special orthogonal group or special linear group of some order , over $\mathbb R$?

For every finite group $G$ , does there exist $n \in \mathbb Z^+$ such that $G$ can be embedded in $SO_n(\mathbb R)$ ? Can every finite group be embedded in $SL_n(\mathbb R)$ for some $n$ ?
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1answer
25 views

Finite Cyclic Group - for each x : x in pow of its order is identity unite

We know that, if G is finite cyclic group, then for each $x\in G$, $$x^{\left|G\right|}=e$$ So I have an easy exercize, to show that $U_8$ isnt cyclic (Without lagrange or something more ...
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$ C_{H}(N) \geq O_{p^{\prime}}(H) $ and $ \overline{H} = H / C_{H}(N) $ is a $ p $-group. Why $ \overline{H} \ltimes N $ is nilpotent?

Suppose $ G $ is a finite group. Let $ H $ be a $ p $-nilpotent normal subgroup of $ G $ and $ N $ is a minimal normal subgroup of $ G $ whose order is divisible by $ p $. Let $ [H , N] \neq 1 $. ...
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22 views

Prove that $kG$ has a central nilpotent element

$k$ is a field of characteristic $p$. $G$ is a finite group, and $p$ divides its order. Book's outline solution If $f=\sum_{x\ni G}x$, then $fy=yf(y\in G)$ and $f^2=|G|f=0$. First of all, ...
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1answer
24 views

To show that H,K are sub-groups of G (U36)

$G\:=U_{36}$ $$H=\left\{\left[x\right]\in U_{36}\::\:x\equiv 1\left(mod4\right)\right\}$$ $$K=\left\{\left[y\right]\in U_{36}\::\:y\equiv 1\left(mod9\right)\right\}$$ I need to show that H,K are ...
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1answer
32 views

Geometric or at least application view of a group with 3 elements?

My teacher asked us to find applications in real life, or ways that a group with 3 elements migth show up in real problems, and the one I gave was about an watch for a planet where the entire day ...
2
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1answer
49 views

Understanding a proof of a lemma to Jordan-Hölder Theorem.

I have difficulty understanding the following lemma. First, how do we know that $M\cap N={1}$, after replacing $M$ and $N$ as in the proof? Next, in the second part of the proof where it says we ...
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1answer
23 views

Center of a maximal subgroup

Is it possible to find a non-abelian group (preferably a finite $p$-group) with the following property? If M is a maximal subgroup of a group (or finite $p$-group) $G$, then $Z(M)\not \le Z(G)$.
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for prime $q\ge p$, integer $m\ge 0$, any group with order $p^2 q^m$ is not simple

I tried the two ways, but both are all failed. First, counting the order of union of all Sylow $p$, $q$-subgroup. Second, group action from orginal group to set of cosets by Sylow subgroup. Is there ...
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1answer
29 views

Find the invariant polynomial space of a finite matrix group.

This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Denote the finite matrix group $\{\pm I_2\}\subset GL(2,k)$ by $C_2$. It is known that the invariant polynomials under $C_2$ ...
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1answer
33 views

Why there exists an element $ y\in G\setminus‎ L $ such that $ K/L_{G} = (\langle y \rangle L_{G})/L_{G} $?

Let $ G $ is finite supersoluble group. Then $ G $ has a chief series. Let $ L $ be a subgroup of $ G $ and $ K/L_{G} $ a chief factor of $ G $ where $ L_{G} $ is the core of $ L $ in $ G $. Since $ G ...
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2answers
33 views

$ N(M\cap H) = M $, $ M \leq NS \leq G $. $ NS = M $ or $ NS = G $ of maximality of $ M $. Let $ G = NS $. Why $ H = S $?

Let $ G $ is a finite group and $ N $ is normal subgroup of $ G $ such that $ G = HN $ for some subgroup $ H $ of $ G $. Suppose $ M $ is a maximal subgroup of $ G $ with $ N \leq M $. Let subgroup $ ...
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0answers
43 views

Why sylow $ p $-subgroups of $ H $ are Sylow $ p $-subgroups of $ G $?

Let $ H $ is a subgroup of $ G $ that $ \vert G : H \vert $ is a $ \pi $-number and there exist a nilpotent subgroup $ K $ of $ G $ that $ G = HK $.then we can let $ K = K_{\pi}K_{\pi^{\prime}}$, that ...
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1answer
26 views

A question about Inner and outer Automorphism

Suppose G is a not abelian group but finite, H$\unlhd$G and K$<$G with H$\bigcap$K=1, then to any k$\in$K, $\phi_k$(h)=$khk^{-1}$ is $\in$Aut(H), my question is: is that possible for some $\phi_k$ ...
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1answer
22 views

Let $ M $ is maximal subgroup of $ G $ and $ P_{G}(M)=G $. Is there $ x\in G\setminus M $ that $ G = \langle x \rangle M = M \langle x \rangle $?

The permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H ...
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39 views

Describe a group $G$ that acts on a set $X$ of 4 elements such that the action of $G$ has 2 orbits.

The motivation of this self made problem is to get a better understanding of Group actions. Say $G$ is a group that acts on a set $X$ of 4 elements such that the action of $G$ has 2 orbits. What ...
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348 views

Is there such a number N such that any group of order N is simple?

Is there such a number N, such that any group of order N is simple?
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50 views

Sylow Theorems to find all groups with order less than or equal to 10

I'm trying to solve the following problem using the Sylow theorems: Determine all groups of order $\leq 10$ up to isomorphism. I know that in particular I have to use the fact that any ...
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2answers
68 views

What if $N$ is not normal in semi-direct product $N\rtimes H$?

In classification of groups of order $n$, we always use semi-direct product. And require $N\unlhd G$, $H<G$ and so on. But what if $N$ is not normal? It seems we can still use ...
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26 views

What is Relator Matrix

At the third section of a paper "Computing second cohomology of finite groups with trivial coefficients" by G. Ellis et al., the authors write Suppose that $<\underline{x}\mid ...
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1answer
48 views

Cayley graphs of groups

(Directed & undirected) Cayley graphs of groups have been studied a lot in the literature. I would like to know the answer to the following questions. Please give your valuable suggestions. Is ...
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1answer
67 views

Norm $\Vert \cdot \Vert$ on the symmetric group $S_n$

If we define a real valued function $\Vert \cdot \Vert$ on the $n^{th}$ order symmetric group $S_n$ satisfying following conditions $$\begin{align} & \|x\|=0\iff x=\omega\,\,\,(\text{identity ...
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1answer
31 views

Set stabilizer of subset vs. set stabilizer of inverse subset

Let $G$ be a finite group and $A\subseteq G$ a subset. The left regular action of $G$ on itself induces a natural action on the powerset of $G$: $$G\times 2^G\rightarrow 2^G,(g,A)\longmapsto ...
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2answers
72 views

The relationship between subfields and subgroups of a finite field.

I am trying to get my head around the structure $GF(p^n)$ when viewed as a vector space of dimension $n$ over $GF(p)$ (mainly the relationship between the additive and multiplicative structures). I'm ...
0
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3answers
45 views

order and cycles of perfect shuffle of 52 cards

This is the shuffle: $$1,2,\cdots,52$$ is turned into $$1,27,2,28,\cdots,26,52$$ when I try to write the cycles of this shuffle, I get a LOT of cycles, for example: $$(2\ 3)(27 \ 2)(26\ ...
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4answers
55 views

How to show that the cycle $(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2)$

I generally do not have any problem multiplying cycles, but I've seen on Wikipedia that $$(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2). $$ I started following the path of $2$ on the right: ...
2
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1answer
106 views

Prove that for a group with even order $2k$, $k$ odd, there is a subgroup $K$ with order $k$ [duplicate]

I'm trying to understand the proof my teacher did: Consider a subgroup $H$ of $G$. If $H$ is not contained in $A_n$, then we can say that there exists at least one permutation in $H$ that is odd ...