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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1answer
54 views

Show that $U(8)$ is Isomorphic to $U(12)$.

Question: Show that $U(8)$ is Isomorphic to $U(12)$ The groups are: $U\left ( 8 \right )=\left \{ 1,3,5,7 \right \}$ $U\left ( 12 \right )=\left \{ 1,5,7,11 \right \}$ I think there is a bit of ...
1
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1answer
18 views

How to find the invariant forms of a finite group

Let $G\subset GL(n,\mathbb{Z})$. I am looking for an algorithm that finds all symmetric matrices $F$ left invariant by G, ie $$g^TFg=F\quad \forall g\in G.$$ I have found lists of these invariants for ...
0
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1answer
28 views

Sylow subgroup of a symmetric group

Consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a $7$-Sylow subgroup of$A_{20}$. Is $H$ cyclic? And is correct the statement which ...
0
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1answer
12 views

Existence of a $G$-invariant metric on a principal bundle

Given a smooth principal $G$-bundle $\pi: P \rightarrow M$, I want to show the existence of connections by showing that $P$ admits $G$-invariant metrics. I was thinking in a kind of averaging ...
2
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1answer
29 views

Not getting how to prove reverse hypothesis.

This is a theorem from Dummit & Foote text- Let $G$ be a group acting on the non-empty set $A$.The relation on $A$ defined by $a \sim b$ iff $a=g.b$ for some $g \in G$ is an ...
1
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1answer
20 views

Converse of Lagrange Theorem for p-groups.

I need clarification on my understanding of Sylow Theorems. Can I say that a finite p-group will have a subgroup for each prime power? If the above is valid, can I say then that p-groups satisfy the ...
2
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0answers
16 views

Simple groups and irreducible characters of degree 3

The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$ and $PSL(2,7)$. That seems to be a result coming from Blichfelt's work on $GL(3,\mathbb{C})$, which I ...
1
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1answer
21 views

Condition in a theorem of Hall

There is a well-celebrated theorem of Hall, which characterizes solvable groups according to the existence of Hall-$\pi$ subgroups. In this theorem, I was wondering whether it can be stated in a ...
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4answers
52 views

If $G$ isn't abelian and $|G|=p^{3}$ then $Z(G)=G'$

Let $G$ a finite group and $G' \subseteq G$ the smallest normal subgroup of $G$ such that $G/G'$ is abelian. Prove that if $G$ isn't abelian and $|G|=p^{3}$ then $Z(G)=G'$ My attempt: If $G$ is not ...
1
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1answer
21 views

If $H$ and $G/H$ are $p$-groups then $G$ is a $p$-group.

Please verify: If $H$ is a $p$-group, then $|H| = p^r$, for some integer $r$. If $G/H$ is a $p$-group, then $|G/H| = p^s$, for some integer $s$. But the cardinality of a quotient set is the index, ...
4
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1answer
40 views

Groups of order $25$

Please verify my solution that there are only two groups of order $25$ up to isomorphism. As $|G|$ is a prime squared, then $G$ is abelian. Since the Theorem of Finite Abelian Groups, $G$ is a direct ...
1
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1answer
23 views

Are there at least $3$ groups of order $16$ that has element of order $8$?

Are there at least $3$ groups of order $16$ that has element of order $8$? I know that probably the simplest way of doing this problem is looking at the element structure of the abelian groups of ...
0
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0answers
25 views

Clarification on a concept involving Hall subgroups

Suppose that $G$ is a finite group and $H\leq G$. If $Q \in$ Hall$_{\pi}(G)$ such that $Q \cap H \in$ Hall$_{\pi}(H)$. Is true that $Q \leq H$ when Hall$_\pi(H)$ $\subseteq$ Hall$_\pi(G)$
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2answers
32 views

If C/A is abelian, then C/B is abelian.

I am trying to prove that if groups A $\lhd$ B $\lhd$ C, A $\lhd$ C, and C/A is abelian, then C/B and B/A are abelian. Clearly, B/A is abelian since it is a subgroup of the abelian group C/A. ...
1
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2answers
52 views

Proof/Reasoning why the sgn function which counts inversions has the following property?

$\mathrm{sgn}(\pi\circ\sigma)=\mathrm{sgn}(\pi)\cdot \mathrm{sgn}(\sigma)$ I am familiar with how to count inversions and any insight for why this formula holds true would be very helpful.
0
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0answers
40 views

Seeing composition table

Let p > 2 be any prime number. The non-zero integers modulo p form a group of order p - 1 with respect to multiplication modulo p. $$ [a][b] = [ab] = ab (mod p) $$ denoted Gp. The identity is ...
1
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1answer
33 views

generators of $\mathbb{Z}_p^*$ are all the elements in $\mathbb{Z}_p^*$?

I know that a finite group with a prime number of elements is cyclic and every element in the group is a generator for the group. Thinking about $(\mathbb{Z}_p^*, \cdot)$ I thought that the order of ...
-1
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3answers
32 views

Group theory finding proper subgroup [closed]

The smallest order for a group to have a non-abelian proper subgroup? I am confused how shall I proceed pls help Thanks in advance
-1
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1answer
20 views

How to find a subgroup of the external direct product $\Bbb{Z}/4\Bbb{Z}\oplus\Bbb{Z}/2\Bbb{Z}$ not of the form $H\oplus K$?

How to find a subgroup of the external direct product $\Bbb{Z}/4\Bbb{Z}\oplus\Bbb{Z}/2\Bbb{Z}$ not of the form $H\oplus K$, where $H$ and $K$ are subgroups of $\Bbb{Z}/4\Bbb{Z}$ and $\Bbb{Z}2/\Bbb{Z}$,...
0
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1answer
25 views

If a normal series has maximal lengh in $G$, then it is a composition series.

A normal series of subgroups of $G$ is a decreasing sequence of subgroups $... \subset G_1 \subset G_0 = G$ where $\subset$ denoted strict inclusion and $\forall i \ \ G_i$ is normal in $G_{...
0
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1answer
22 views

Showing that $\phi$ is a homomorphism

$G=\mathbb{Z}^2$ is a group with product $(a,b)\cdot(c,d)=(a+c,(-1)^cb+d)$. Show that the image $\phi: G \to D_{10}$ with $(a,b) \mapsto s^ar^b$ is a homomorphism ($D_{10}$ is the dihedral group of ...
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0answers
23 views

Irreducible complex continuous unitary finite dimensional representations of SO(2)

I have to find all continuous finite dimensional complex Irreducible and unitary representations of $SO(2)$. I know that every element of $SO (2)$ can be written as $exp(J \theta ) $, where $\theta$ ...
3
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1answer
32 views

What is the kernel of $\phi$?

Let $\phi: \mathbb{C}^* \to \mathbb{R}^*$ with $z \mapsto |z|$ be a homomorphism. What is the kernel of this homomorphism? We know the identity in $\mathbb{R}^*$ is $1$. So we need to find the ...
7
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3answers
318 views

Prove a group of order 12 must have an element of order 2 [duplicate]

Question: Prove that a group of order 12 must have an element of order 2. I believe I've made great stride in my attempt. By corollary to Lagrange's theorem, the order of any element $g$ in a group ...
0
votes
1answer
35 views

Is there an onto homomorphism $S_4\to S_3$ [duplicate]

Prove or disprove that there is an onto homomorphism from $S_4\to S_3$ where $S_n$ is the symetric group of order $n!$. after long time of searching, I finally success but i just manually tried to ...
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1answer
27 views

Need help for the example on conjugation.

This is an example from Dummit & Foote text, i've some queries in this- If $|G|>1$, then unlike action by left multiplication,$G$ does not act tranistively on itself by conjugation because {1}...
2
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0answers
28 views

Permutations associated to a transversal and Cayley theorem

Let $G$ be a finite group with $H\le G$ and $T$ a right transversal of $H$ in $G$. $G$ acts on itself by left multiplication and so we can consider $G\le \mathfrak{S}_G$. Let $g\in G$. The permutation ...
0
votes
0answers
32 views

Explanation to proof of Fermat's little theorem

Fermat's theorem: For every $a \in \mathbb{Z}$, and every $p \in \mathbb{P}$. Then, $a^{p}\left ( modp \right )\equiv a\left ( modp \right )$ Proof: By the division algorithm, $a=pm+r,\exists 0\...
0
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0answers
28 views

Suppose φ : G → G is a group homomorphism, and H is the image of G under φ. If G is of polynomially bounded growth, show that H is too.

Suppose φ : G → G is a group homomorphism, and H is the image of G under φ. If G is of polynomially bounded growth, show that H is too. [the question is above as well as in the title] how would I ...
1
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1answer
24 views

A natural example of f.g. group, with uncountably many normal subgroups, but is not SQ-universal

I am looking for "natural" examples of a finitely generated group which have uncountably many normal subgroups but are not SQ-universal. A group $G$ is SQ-universal if for any countable group $H$, $H$ ...
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0answers
26 views

Prove that FG is of exponential growth where F is any field and G is the free group on {x_1, . . . , x_n}. [closed]

Prove that FG is of exponential growth where F is any field and G is the free group on {x_1, . . . , x_n}. where FG is: Given a group G and a field F, we can combine them into a group algebra. Setwise, ...
0
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3answers
78 views

Why is this set not a group?

So the group $Z_8 - \{0\}$ does not form a group under multiplication modulo $8$. The reason being is because its not closed as $2 \cdot 4 = 0$. I understand what closure is. If $a$ and $b$ are in ...
2
votes
2answers
42 views

$G \cong H$ and $G$ is simple. Then $H$ is simple as well.

I think it must be true. Yet I have no rigorous proof for that. So, what I need to prove is that "group being simple" is invariant under isomorphism. That if $G \cong H$, then either both are simple ...
2
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0answers
60 views

How can we show $gnu(8892)=gnu(9324)$ by hand?

With GAP it can be verified immediately that there are $303$ groups of order $8892=2^2\cdot 3^2\cdot 13\cdot19$ and also $303$ groups of order $9324=2^2\cdot3^2\cdot 7\cdot 37$. I do not expect that ...
0
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1answer
18 views

Show how (0,(12)) and (1,(12)) are in different conjugacy classes.

$S_3=${${(1),(12),(13),(23),(123),(132)}$}. $\mathbb Z_2=${${0,1}$}. $\mathbb Z_2$$\oplus$$S_3$={$(0,(1)),(0,(12)),(0,(13)),(0,(23)),(0,(123)),(0,(132)),(1,(1)),(1,(12)),(1,(13)),(1,(23)),(1,(123)),(...
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2answers
38 views

$\mathbb{Z}$ has no composition series. Need an assistance in some questions.

Please, read the whole post before trying to answer. Remark: here $\subset$ means "strict inclusion". I need to prove that the group $\mathbb{Z}$ has no composition series. That is, no normal series ...
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0answers
16 views

Let $G=AB$ where $(|A|,|B|)=1$ and $V$ be an $\mathbb{F}[G]$ module.

Under these assumptions it is a well-known fact that if $V_A$ and $V_B$ are faithful ($V_A$ denotes $V$ as an $\mathbb{F}[A]$-module) then $V$ is also faithful. Clearly if $V_A$ and $V_B$ is ...
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0answers
41 views

Class-equation of $\mathbb Z_2$ $\oplus$ $S_3$.

$S_3=${${(1),(12),(13),(23),(123),(132)}$}. $\mathbb Z_2=${${0,1}$}. $\mathbb Z_2$$\oplus$$S_3$={$(0,(1)),(0,(12)),(0,(13)),(0,(23)),(0,(123)),(0,(132)),(1,(1)),(1,(12)),(1,(13)),(1,(23)),(1,(123)),(...
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0answers
16 views

Intersection of the kernel of the irreducible characters determinants

Let $G$ be a finite group. It is easy to show that $G'\le \bigcap_{\chi\in Irr(G)}Kerdet\chi$. Is there equality ? This question arises from the remarkable equalities $\bigcap_{\chi\in Irr(G)}Ker\...
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0answers
43 views

Determinant of a character

let two characters $\chi$ and $\vartheta$ of a finite group $G$ (assumed to be non-null). Let $\mathfrak{X}$ and $\mathfrak{Y}$ be representations of $G$ affording respectively $\chi$ and $\vartheta$ ...
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2answers
42 views

Are the groups $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices and $O(n,\mathbb R)$ isomorphic?

Let $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices ; Are $G$ and $O(n,\mathbb R)$ isomorphic ?
2
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2answers
26 views

A simple group such that $[G:H]=n$ can be embedded into $A_n$

Let $G$ be a finite simple group and $H$ be a proper subgroup of $G$ such that $|G:H|=n$. Then, how do I prove that $G$ can be embeded into $A_n$? I can prove that $G$ can be embedded into $S_n$ ...
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0answers
40 views

Recent advancement in Haar measure

From my personal interest I have studied Haar Measure and the related concept of group theory on my own. However due to the lack of an authoritative source it is not getting possible for me to know ...
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2answers
29 views

Why if $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic? [duplicate]

In the notes I'm studying from ( again =) ) I read: If $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic Could someone give me a justification for ...
2
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2answers
19 views

Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
2
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1answer
45 views

On the maximum number of Sylow subgroups

I have some doubts regarding the first sentences below the fourth case in the more elementary solution by Prof. Samuel to problem 11856 of the Monthly. Here you have a screenshot of the relevant part ...
0
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2answers
31 views

$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?

Let $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then is it true that $G=HK$ ? ( I know that the fact is true if $p=2$ ...
1
vote
1answer
53 views

A proposition about free product

My question is about a proposition that I found on Munkres book: Topology (page 419). My definitions are based on this book. I give you the definition of free product that he uses. $\textbf {...
2
votes
3answers
29 views

Number of orbits with Burnside's lemma

We color a equilateral triangle by coloring each edge with one of $k \geq 1$ colors. Find a formula for the number of orbits under the action of $D_6$, the dihedral group of $6$ elements, on the ...
9
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1answer
84 views

Does $\forall v ( T_1 v = 0 \lor T_2 v = 0 \lor \dots \lor T_n v =0 )$ imply $T_1 = 0 \lor T_2 = 0 \lor \dots \lor T_n = 0$?

Let $V$ and $W$ be vector spaces and $T_1$, $T_2$, $\dots$, $T_n$ be linear transformations from $V$ to $W$, such that for every $v$ in $V$, either $T_1 v = 0$, $T_2 v = 0$, $\dots$ or $T_n v = 0$. ...