A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Show that there has to be an orbit with at least $3$ elements

Let $H \leq A_4$ be a subgroup of $6$ elements. $H$ acts on $X=\{1,2,3,4\}$ by $\sigma \cdot i = \sigma(i)$. Show that for $H$ to act on $X$ there has to be an orbit with at least $3$ elements. ...
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43 views

Order of subgroup generated by two cyclic subgroups in $S_6$.

Let $S_6$ be the symmetric group, and $\alpha=(13456)$ and $\beta=(132)$ be its two permutations. How can we find the order of the subgroup generated by $\alpha$ and $\beta$. SOl: $\alpha^5$=...
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32 views

Properties of non-abelian characters

I'm looking for some (short of) non-abelian generalization of the following result: Let $G$ be a finite abelian group and let $f$ be a function on $G$ with values in some field of characteristic ...
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2answers
34 views

Is there a theorem stating that disjoint cycles generate distinct elements?

If we have a group $H=\langle (12345),(678) \rangle$, it's obvious that $|H|=|(12345)|\cdot |(678)|=15$, because the cycles are disjoint. Is there some theorem stating this?
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1answer
49 views

Representation of of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$

Certain part in my textbook implies that a representation of $SO(3)$ in the vector space $V = \mathbb C^{2S+1}$, where $S \in \mathbb Z$, is possible. I am trying to find a path that leads to this ...
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26 views

Alternative proof : Group algebra contains all irreducible G-modules.

It is well-known that the group algebra $F[G]$ is a direct sum of irreducible $G$-modules. The proof in my text book is as follows Write $F[G] = \oplus_{i=1}^n V_i$, where $V_i$ is a set of ...
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1answer
41 views

quotient group of finite fields

I could not define the elements of the quotient group $F^*_{q^{d}} / F^*_{q}$. Let $F^*_{2^{4}}$={$1, a, a^2, a^3, a^4=a+1, a^5=a^2+a,...a^{14}=a^3+1,a^{15}=1$} s.t. $ a^4+a+1=0$ and $F^...
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26 views

Existence of open subgroup extending a smaller one

Let $G$ be an abelian topological group and $H \subseteq G$ a dense subgroup (equipped with the subset topology). Furthermore let $V \subseteq H$ be a subgroup that is open in $H$. Does there exist a ...
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1answer
31 views

Proof of a theorem on Reflection Groups

I am reading the book Finite Reflection Groups by Grove and Benson. I didn't understand the following proof. See $(a_1,t)$. What is $t$ here? Then Why the inequality $(r,t)-2(r,r_{i_1})(r_{i_1},t)&...
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12 views

Class Preserving Autmorphisms [closed]

What are Camina p groups? What special properties do Camina groups of Class 2 have over those of class 3?
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25 views

Groups with order a product of unrelated distinct primes

Consider a group of size $n$, where $n$ is the product of distinct unrelated primes (two primes $p$ and $q$ are unrelated if $q \nmid (p-1)$ and $p \nmid (q-1)$). The claim is that there is only one ...
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1answer
60 views

If $|G/H|=4$ then $G$ is union of three proper subgroups

Let $G$ be a finite group and $N$ a normal subgroup of $G$ of index $4$. Prove that $G$ has a subgroup of index $2$ (hence normal) and if $G/H$ is not cyclic then $G$ is equal to the union of three ...
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1answer
46 views

Minimum number of elements in group

Suppose we have $G=\langle(123),(456),(14)(25)(36)\rangle$ a subgroup of $S_6$ Am I correct in saying that the minimum number of elements in $G$ equals the product of the orders of the elements of ...
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30 views

Irreducible action of a group

Suppose that $G = HN$ is a finite solvable group where $N$ is a minimal normal subgroup of $G$ Does $H$ act irreducibly on $N$? I know that $N$ is an elementary abelian $p$-group of $G$. I need to ...
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1answer
65 views

“Concrete” Realisations of non-abelian finite groups

Many commutative groups could be imagined as something "concrete", for example $\mathbb N$ as an abstraction of operations on sets of objects, and from this $\mathbb Z, \mathbb Q$ and $\mathbb R$ are ...
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74 views

Sylow $p$-subgroup of a finite group

I know: Let $P$ be a Sylow $p$-subgroup of a finite group $G$.If $N$ is normal in $G$, then $P \cap N$ is a Sylow $p$-subgrup of $N$. But if $N$ is not normal in $G$ , there is also the issue? ...
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14 views

Show that the centralizer is generated by these $3$ elements

$\sigma = (123)(456)$. Show that $C_{S_6}(\sigma)=\langle(123),(456),(14)(25)(36)\rangle$. So we know that there are $40$ elements conjugate with $(123)(456)$ in $S_6$. Then it follows that $|C_{...
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38 views

Number of elements set of generators

Suppose $G$ is a subgroup of $S_6$ generated by $G=<(123),(456),(14)(25)(36)>$. Is it safe to assume that products of distinct generating elements are also distinct? What I mean is, can we be ...
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1answer
56 views

reference for classifying groups of order $p^2q^2$

In a previous question I asked about the number and structure of groups of order $p^2q^2$ where $p,q$ are primes and with the help of Prof. Derek Holt I understand it now (see here non-abelian groups ...
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1answer
29 views

What is the centralizer of $(123)(456)$ in $S_6$?

Given that $\sigma = (123)(456)$. Compute $C_{S_6}(\sigma)$ (instead of writing out all the elements, write down the elements that generate the centralizer). If $\sigma$ were an $m$-cycle we chould ...
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1answer
29 views

The representation of $SU(2)$ as a polynomial function on $\mathbb C^2$

Let $A$ element of $SU(2)$ and $p$ a polynomial function of fixed degree $l$ on $\mathbb C^2$ (in other words, $p \in P_l(\mathbb C^2)$), then the polynomial representation of $A$ in $P_l(\mathbb C^2)$...
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2answers
44 views

Group of order $54$ has normal sugroup of order $27.$

Let $G$ be a group of order $54$. Prove that there exists a normal subgroup of order $27.$ Is this normal subgroup unique? Thoughts. Since $27$ divides $54$, by Lagrange's theorem we can not exclude ...
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3answers
53 views

G acts on X transitively, then there exists some element that does not have any fixed points

Let $X$ be a transitive $G$-set. ($G$ acts on $X$ transitively.) If $X$ is finite and has at least two elements, show that there is some element $g$ $\in$ G which does not have any fixed points; that ...
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1answer
34 views

The number of different G-actions on X [closed]

Let $X$ $=$ $\{$$1$, $2$, $3$$\}$ and $G$ $=$ $\mathbb Z_2$. How many different G-actions are there on $X$? Just learned group action. Need some hint on this one. Thanks.
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36 views

Number of elements in $S_6$ conjugate to $(123)(456)$

Find the number of elements in $S_6$ conjugate to $(123)(456)$ I know we're only looking at elements in $S_6$ with the same cycle type as $(123)(456)$ (two 3-cycles). So we have the following: $$\...
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28 views

Dirchlet region for the Hecke Triangle group

Let $G_n$ for $n>2$ be the subgroup of $SL_2(\Bbb R)$ generated by $$ \begin{bmatrix} 0 & -1\\ 1 & 0 \\ \end{bmatrix} \ \text{and} \ \begin{...
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1answer
45 views

Classifying groups of order 6

I'm trying to proof that if a group $G$ has order $6$, then it is either $\mathbb{Z}_{6}$ or $S_{3}$. I know that there are a lot of solutions to this on the internet, but I want to know why I found ...
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1answer
27 views

Abelian fitting subgroup of a solvable groups

Let $G$ be a finite solvable groups and $F(G)$ be a Sylow $p$-subgroup of $G$, for some prime $p$ such that $F(G)$ is Abelian. I want to know what things we can say about $F(G)$ in this case?
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26 views

Subgroup of $\left(GL_2\left(R\right),\:\cdot \right)$

Is $t\left(GL_2\left(\mathbb{R}\right)\right)=\left\{x\in GL_2\left(\mathbb{R}\right)|\:ord\left(x\right)\:<\:\infty \right\}$ a subgroup of $\left(GL_2\left(\mathbb{R}\right),\:\cdot \right)$ ? ...
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35 views

Find all subgroups of order $4$ in $Z_4 \oplus Z_4.$

Find all subgroups of order $4$ in $Z_4 \oplus Z_4.$ My attempt We begin to find all elements of order $4$ in $Z_4 \oplus Z_4.$ First attempt is to find all the cyclic subgroups of order $4.$ We want ...
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1answer
60 views

Let $S$ be a subset of $G$ such that for any $x,y \in G$ , $xS ; yS$ are either disjoint or equal , then is $S$ a left coset of some subgroup of $G$?

Let $G$ be a group , for any subset $S$ of $G$ and $g \in G$ , let $gS:=\{gs:s\in S\}$ ; Now suppose $S$ is a subset of $G$ such that for any $x,y \in G$ , either $xS \cap y S = \varnothing$ or else $...
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1answer
66 views

$G$ be a finite simple group , then every element of $G$ can be written as a product of $n$-th powers of elements of $G$?

Let $G$ be a finite simple group , let $n$ be a positive integer such that not all $n$-th powers of elements of $G$ are identity , then is it true that every element of $G$ can be written as a ...
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3answers
100 views

Property of odd ordered elements of a Group

I'm (slowly) working my way through "Abstract Algebra" by Dummit and Foote. In the first set of exercises on group theory, the following question is posed: "Let $G$ be a finite group and let $x$ be ...
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26 views

Relation between subgroups of a cyclic group of order $p^n.$

Let $G$ be a finite cyclic group of order $p^n$, where $p$ is a prime and $n \geq 0 .$ If $H \& K$ are subgroups of $G$, then show that either $H \subset K$ or $K \subset H.$ My attempt Let $G$ ...
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1answer
49 views

Finite {2.3}-group with 4 Sylow 3-subgroup

Let $G$ be a finite {$2$,$3$}-group, the number of Sylow $3$-subgroups of $G$ be $4$, and a Sylow $2$-subgroup of $G$ be normal in $G$. Let $N$ be the kernel of the conjugation action of $G$ on its ...
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1answer
41 views

Cohomology $H^1(G{L_n}({\mathbb{Z}}/{2\mathbb{Z}}),({\mathbb{Z}}/{2\mathbb{Z}})^n))$

$\newcommand{\Z}{\mathbb{Z}}$ In my thesis I had a problem that could be solved by proving that $H^1(GL_n(\Z/2\Z),(\Z/2\Z)^n)$ is trivial for all $n\geq 2$. This is something my supervisor said, but I ...
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39 views

adding a relation to a group

We start with a group $G$ such that there is an abelian subgroup $\mathbb{Z}^3(e,f,g) \subset G$. Assume we have $G/\left\{ g = 1 \right\} \cong \mathbb{Z}^2(e,f)$. What can one say about $G$, except ...
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1answer
107 views

Primitive Wreath Product action

I am a little confused about the primitive action of the wreath product as when to use the inverse and whether to use left or right action. Let $H, K$ be groups and $K$ acts on $\Delta$, the wreath ...
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48 views

If $\sigma \in S_n$ has order some prime $p$, then is $|\{1 \le i \le n : \sigma(i)=i\}|\equiv n \pmod p$? [closed]

Let $\sigma \in S_n$ be such that $o(\sigma)=p$ (some prime). Then is it true that $$|\{1 \le i \le n : \sigma(i)=i\}|\equiv n \pmod p\ ?$$
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Prove that the logarithm function is an isomorphism of totally ordered abeligan groups of $\mathbb{P}$ onto the additive group $(\mathbb{R}, +)$

The book of Abstract Algebra by Antonio Grillet, gives me the following example in the part of Valuations The logarithm function is an isomorphism of totally ordered abeligan groups of $\mathbb{P}$...
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3answers
61 views

If $\phi: G \rightarrow H$ is a group homomorphism, $N \vartriangleleft G$, then $G/N \cong \phi(G)/\phi(N)$

I wish to prove whether this is true or false. If $\phi: G \rightarrow H$ is a group homomorphism, $N \vartriangleleft G$, then $G/N \cong \phi(G)/\phi(N)$. I'm not even sure if $N$ being normal ...
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1answer
47 views

Subgroups of a Galois Group [on hold]

I've just started studying Galois Theory and I'm having a litte trouble with the following exercise: Find all the subgroups of $\operatorname{Gal}\big(X^4-X^2 -2\ ;\mathbb{Q}\big)$. Which of the ...
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3answers
51 views

If a Lie Algebra is solvable, is the corresponding Lie group solvable in the group theoretic sense?

I just started working with Lie Algebras with a professor. The way we defined them is probably the standard way; treat Lie Algebras as tangent spaces at the identity of the Lie Group. Now, consider ...
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1answer
67 views

Name of the theorem: If $p^k$ divides $|G|$, then $G$ has a subgroup of order $p^k$?

Note: I am not asking for a proof of this theorem or any other theorem or help with a mathematical problem. This question is a reference request. I use the following well-known and somewhat-easy-to-...
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23 views

An exercise about the characterization of procyclic groups

I need a hint for the following exercise. Here $\mathbb{Z}_{p}$ denotes the $p$-adic integers. I can use the fact that $\hat{\mathbb{Z}}\cong\prod_{p}\mathbb{Z}_{p}$. Exercise. Let $G$ be a ...
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64 views

An example of a non-abelian group of order $p^3$

Let $p$ be prime and consider the set of all matrices $$\begin{bmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{bmatrix}$$ where $a,b,c \in \mathbb{Z_p}$. This set forms a non-...
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49 views

Show that $\mathcal D_{12}\cong C_2\times \mathfrak S_3$ [duplicate]

Let $\mathcal D_{12}=\left<a,b\mid a^6=b^2=1,b^{-1}ab=a^{-1}\right>$, $\mathfrak S_3=\left<(12),(123)\right>$ and $C_2=\left<g\mid g^2=1\right>$. I would like to show that $\mathcal ...
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1answer
19 views

A necessary and sufficient condition for finite normal subgroups using group actions.

For $H$ a finite subgroup of $G$, the mapping $$H\times H \rightarrow G: (h,h') \mapsto hxh'^{-1}$$ defines an action of $H\times H$ on $G$. Show that $H \lhd G$ if and only if every orbit of this ...
3
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1answer
40 views

Identity element of matrix group

Suppose we have the following: $$G=\Bigg\{ \left[ \begin{array}{ c c } \overline{a} & \overline{b} \\ \overline{0} & \overline{c} \end{array} \right] \Bigg|\overline{a},\overline{...
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1answer
28 views

Automorphisms of Free Groups of Finite Order

The automorphism group of a free group (of finite rank) is known (see this). The group is infinite, if the number of generators is at least $2$, since there are automorphisms of the form $x_i\mapsto ...