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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Relation between torsion in torsion free of covariant derivative and torsion free group

Is there a relationship between "torsion free" of covariant derivatives and the torsion free group? Or is this just coincidence that people use the term "torsion free" here? It is in general required ...
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24 views

Presentation of the symmetric group of 5 symbols.

I am trying to write the presentation of the symmetric group $S_{5}$. We know that $S_{5}$ is generated by $a=(1,2)$ and $b=(1,2,3,4,5)$. Using this I am trying to write presentation of $S_{5}$. My ...
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29 views

For any $A, B \in SL(2, F)$, does knowing $\operatorname{tr}A$, $\operatorname{tr}B$, and $\operatorname{tr}AB$ specify $A$ and $B$?

In title, $F$ denotes a field. Does knowing the trace of two matrices and their product specify those two matrices? Up to some equivalence, perhaps?
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Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle$.

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle $. I would like someone to check my solution. First of i will prove that $G$ ...
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The point of writting this isomorphism theorem like this?

In group theory there is this isomorphism theorem that doesn't seem to give any special information the way it is written. Let $T\unlhd G$ and let $S\leq G$ then $\frac{S}{S\cap T}\cong ...
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29 views

Determine the galois group of $x^5+sx^3+t$

im trying to show that the galois group of $x^5+sx^3+t$ over $\mathbb{Q(s,t)}$ is $S_5$. By just looking at the discriminant, it has to be $S_5$ or $F_{20}$. I know i could distinguish between those 2 ...
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1answer
58 views

Necessity of being well-defined in Group Homomorphism?

In Group Theory, homomorphism is isomorphism when we no longer restrict to bijective map; do we still need that map to be well defined in homomorphism (like in isomorphism) or homomorphism can be ...
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24 views

How many permutations cover alternating/reverse alternating permutations?

Given integers $1$ through $2n$, let $S$ be set of ordering of integers that respect even alternating or reverse alternating permutations (https://en.wikipedia.org/wiki/Alternating_permutation) up to ...
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101 views

How to prove the group $G$ is abelian?

Question: Assume $G$ is a group of order $pq$, where $p$ and $q$ are primes (not necessarily distinct) with $p\leqslant q$. If $p$ does not divide $q-1$, then $G$ is Abelian. I know that if the ...
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1answer
46 views

Prove or disprove: $(\mathbb{Z}^*, \cdot)$ and/or $(\mathbb{Z}^*, \div)$ is a group.

I am teaching myself information about groups, but don't really understand how to work through this problem. Here is what I have been thinking so far (please note that I do not need to work through ...
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27 views

Are permutation group block only defined in the context of finite sets?

From Dummit and Foote (emphasis mine): Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ s.t. $\forall \sigma\in G:$ either $\sigma(B)\cap B ...
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30 views

A question about normal subgroups of nilpotent group

Assume G is a nilpotent group and to any n dividing $|G|$, if there is always a normal subgroup of G with order n?
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59 views

Show that $f$ is a homomorphism.

There is a group $G$ of order $p^3$, where $p>2$. Show that $f:G\rightarrow Z(G) $ with $f(x)=x^p$ is a homomorphism. My attempt: Case a): Suppose $|Z(G)|=p^3$. Then $G=Z(G)$, so $G$ is abelian, ...
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28 views

Where do I use that $G$ is a permuation group?

This is about question $4.1.7$ from Dummit and Foote, and also related to my previous question. The question is (summarised a bit): Let $G$ be a transitive permutation group on a finite set $A$. ...
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1answer
34 views

How to read $[G:N]$?

For a group $G$ and a normal subgroup $N$ of $G$, the quotient group of $N$ in $G$ is written $G/N$. I could find from this link how to read $G/N$ ("$G$ modulo $N$" or "$G$ mod $N$"), but I couldn't ...
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31 views

How to prove the theorem on group algebra of permutation group [on hold]

How to prove the following theorem: If $t$ is a vector in group algebra of permutation group, then $\cal{y}t\cal{y}=\lambda_t \cal{y}$, where $\cal{}y$ is the Young operator of permutation group and ...
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91 views

Subring of $M_7(\mathbb{Z}_2)$ isomorphic to $\mathbb{F}_{128}$?

Let $A \subset M_7(\mathbb{Z}_2)$ be a subring such that no proper nonzero subgroup $V \subset \mathbb{Z}_2^7$ is invariant under all matrices in $A$. I suspect that $A \cong \mathbb{F}_{128}$, but ...
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2answers
84 views

Algebraic proof that a set generated by irrational rotations is dense in $S^1$.

This is exercise 1.9 in Lie Groups, Lie Algebras and Representations - Hall. Suppose $a$ is an irrational real number. Show that the set $E_a$ of the numbers of the form $e^{2\pi i n a}$, $n \in ...
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3answers
61 views

Defining a coproduct in $\mathsf{Grp}$ using group presentations

I've encountered this exercise in Aluffi's Algebra: Chapter 0. It might be helpful to say that the book doesn't introduce functors at this stage,, and that the book defines a presentation of a group ...
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44 views

Prove that $(\mathbb{Z}_n , +)$, the integers (mod $n$) under addition, is a group.

Prove that $(\mathbb{Z}_n , +)$, the integers (mod $n$) under addition, is a group. To show that this is a group, I know I need to show three things (in our text, we do not need to show that addition ...
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36 views

Difference between “$G$ acts on $A$” and “G is a permutation group on $A$ (i.e. $G\leq S_A$)”

This question is inspired by questions $4.1.1$ and $4.1.2$ of Dummit and Foote. The hypothesis for the first question is formulated as: "Let $G$ act on the set $A$", and the hypothesis for the second ...
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First appearance of modern definition of a group [migrated]

What is the first appearance in print of the modern definition of an abstract group? To qualify, it should be a formal definition, contain the word "elements" (so Burnside's 1897 restriction to ...
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33 views

Commutator subgroup of general linear group [on hold]

Let $G$ and $S$ be the group of all invertible $n\times n$ matrices and invertible matrices with determinant $1$ of the same order respectively over the field of real numbers. Prove that $S$ is ...
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topic between algebra and geometry [on hold]

I have to do an exam on Differential Geometry and my teacher wants that I prepare a choosen topic, outside lectures program, that I will talk about at the oral part of the exam. I am interested in ...
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38 views

Number of conjugacy classes of nonabelian group of order $pq$.

The problem asks to show that a nonabelian group of order $pq$ has $p+\frac{q-1}{p}$ conjugacy classes. I have shown a. $p$ divides $q-1$, b. $|Z(G)| = 1$, Now I'm using the class equation to ...
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1answer
30 views

H subgroup of G such that H=Inn(G)=Z(G) [on hold]

Let $G$ be a group such that the condition in the title is fulfilled. What can be said about $H$? Is it finite? Cyclic? Many thanks in advance.
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9 views

Weyl (anti-)invariant differential operators on spheres

The permutation group $S_n$ acts as via Weyl reflections of $A_{n-1}$ on $R^{n-1}$ und thus on the sphere $S^{n-2}$. On this sphere, we have a natural action of $SO(n-1)$ generated by the angular ...
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1answer
28 views

The set of nonneg integral powers of $2$ is a group under $\max(a,b)/\min(a,b)$ [on hold]

Let $X$ be the set of all nonnegative integeral powers of $2$. Prove or disprove: $X$ is a group under the operation $a . b = \max(a,b)/\min(a,b)$
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Why is $\phi : \operatorname{Hom}(\mathbb{Z},G) \to G$ given by $ f \mapsto f(1)$ surjective?

I was working on showing $\operatorname{Hom}(\mathbb{Z},G) \cong G$ for $G$ abelian. The proposed map given by evaluating a given $f \in \operatorname{Hom}(\mathbb{Z},G)$ at $1$ is easily seem to be a ...
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2answers
40 views

In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian

Let $G$ be a group. If for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then phow can I prove that $G$ is abelian?
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42 views

In a group $G$, prove the following result

Let $G$ be a group in which $a^5=e$ and $aba^{-1}=b^m$ for some positive integer $m$, and some $a,b\in G$. Then prove that $b^{m^5-1}=e$. Progress $$aba^{-1}=b^m\Rightarrow ab^ma^{-1}=b^{m^2}$$ ...
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28 views

Show that in a finite abelian group $G=\{a_1,\dots, a_n\}$, and $x=a_1\dots a_n$, then $x^2=e$ [duplicate]

Let $G=\{a_1, a_2, \dots, a_n\}$ be a finite abelian group and $x=a_1a_2\dots a_n$. Then show that $x^2=e$. Let $a_i\in G$, then $a^{-1}_i\in G$. Suppose $a_i^{-1}=a_j$. How can show the required ...
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31 views

Possible orders of the elements of the alternating group $A_n$

What are the possible orders of the elements of the alternating group $A_n$?
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+50

Why Composition and Dihedral Group have reverse order of operation?

NOTE - I didn't receive any answer in here and I think because my first post is not clear, so I entirely made another example: $K={\{id,r^2,r^4,s,r^2s,r^4s}\}$ is a proper subgroup of the dihedral ...
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1answer
37 views

Prove that $H\times K \cong K\times H$

According to the book: Let $G$ be the internal direct product of subgroups $H$ and $K$. Then $G$ is isomorphic to $H\times K$. From that it results $H\times K \cong K\times H$. Is there any ...
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Discrete series representations of the group SO(2n,1)

I am interested in discrete series representation of the group SO(2n, 1). Can someone recommend me a paper or a book about it?
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On the definition of Internal Direct Products

Let $G$ be a group with subgroups $H$ and $K$ satisfying the following conditions. • $G=HK={\{hk:h∈H,k∈K}\}$; • $H∩K={\{e}\}$; • $hk=kh\ \text {for all}\ k∈K \text {and}\ h∈H$. My question is: ...
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1answer
70 views

Intuitively and Mathematically Understanding the Order of Actions in Permutation GP vs in Dihereal GP

I define $r$ to be one rotation clockwise, and s to be reflection on the 'horizontal' line (see the figure). So I can make these bijections: (in clockwise order) $$\begin{align*} 1,2,3,4,5,6 ...
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1answer
75 views

4th Isomorphism Theorem applied to normalizers

I'm reading a proof showing a proper subgroup Q of p-group P is contained in it's normalizer. It applies the 4th Isomorphism Theorem to assert $\frac{Q}{Z(P)} < ...
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Explaining elementary arithmetic in terms of group theory

It is possible to explain elementary arithmetic in terms of group theory? Addition and subtraction seem to be fine using $(\mathbb{R},+)$ but when it comes to multiplication and division it does not ...
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Let $ O_{p^{\prime}}(G/A) = T/A $, Why $ T \leq F $ and $ [A , T]=? $

Let $ G $ be a soluble group and $ A $ be a minimal normal subgroup of $ G $,where $ A $ is an elementary abelian group of prime power order. Let each chief factor of $ G/A $ has order $ 4 $ or a ...
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Is 1 always an element in multiplicative group?

Let $\mathbb{G}_T$ be a multiplicative group. Is 1 $\in \mathbb{G}_T$ ? I think its true, because if $a \in \mathbb{G}_T$ then $a^{-1} \in \mathbb{G}_T$, So $aa^{-1} =1 \in \mathbb{G}_T$. Is the ...
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Prove $K\cong S_3$

In the book Abstract Algebra by Thomas W. Judson, Ch. 9 Example 13 - Example 13. The dihedral group $D_6$ is an internal direct product of its two subgroups $H={\{id,r^3}\}$ and ...
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Let $ F = VN $ that $ V \cap N = 1 $ . Let $ L = N_{G}(V) $. $ (\vert N \vert , \vert F/N \vert) = $?

Let $ G $ be a soluble group and $ A $ be a minimal normal subgroup of $ G $,where $ A $ is an elementary abelian group of prime power order. Let each chief factor of $ G/A $ has order $ 4 $ or a ...
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50 views

Characterisation of the squares of the symmetric group

I found out that for $n\le 4$ we have $S_n^2=A_n$ with $G^2$ defined by $$G^2:=\{g^2 \mid g\in G\}$$ for any group $G$. Surely we have $S_n^2\subseteq A_n$ for all $n\in\mathbb N$. Is there a ...
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65 views

Is the subset of squares of a group a subgroup?

Let $G$ be a group and $$S:=\{g^2 \mid g\in G\}$$ the subset of all squares of $G$. Is $S$ then a subgroup? I would say no since I don't see why $g^2h^2$ should be a square in the non-abelian case. ...
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31 views

The class equation of the octahedral group

I know that the class equation of the octahedral group is this: $$1 + 8 + 6 + 6 + 3$$ I think the $8$ stands for the $8$ vertices, the $6$ could be $6$ faces and $6$ pairs of edges. Then what is the ...
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A question about primitive idempotent of group algebra [on hold]

How to prove: $e_j$ is primitive idempotent of group algebra $\cal{L}$ iff $\forall\ t\in \cal{L}\ $, $e_j^2=e_j$ and $e_j t e_j=\lambda_te_j$. Or in which book can I find the proof.
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infinite-order elements of $Out(\widehat{F_2})$

Let $\widehat{F_2}$ be the pro-$\ell$ completion of the free group of rank 2, where $\ell$ is some prime. Every outer automorphism of $F_2$ induces an outer automorphism of $\widehat{F_2}$, hence an ...