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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103 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
62 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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0answers
25 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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2answers
112 views

More Symmetric than the symmetric groups?

So I was considering the following question. Is there a group of size n! (or less), that contains a larger number of subgroups than $$S_n$$? In thinking about this problem one obvious contender that ...
5
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3answers
344 views

Can one construct a “Cayley diagram” that lacks only an inverse?

My group theory text asks for an example of a Cayley-like diagram that exhibits all the properties of a group except (only) that at least some elements lack an inverse. Is it possible to construct ...
6
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1answer
121 views

Extension of group with Ext$^{1} (A, B) = 0.$

Are there any infinite torsion free abelian groups $A$ and $B,$ with $A$ is not projective and $B$ is not divisible but $$\text{Ext} ^{1}(A, B) = 0.$$ Thanks
1
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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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0answers
31 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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1answer
31 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?
5
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2answers
62 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, ...
2
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1answer
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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32 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 ...
4
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2answers
162 views

Definitions in a Theorem of Lang

I'm trying to understand the following theorem due to Serge Lang (Algebraic Groups over Finite Fields, 1956, Theorem 2): Let $p$ be a prime, and let $k$ be a finite field of $q=p^n$ elements. Let ...
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0answers
29 views

A question about primitive idempotent of group algebra [closed]

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.
3
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1answer
99 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 ...
0
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0answers
33 views

Commutator subgroup of general linear group [closed]

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 ...
2
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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 ...
4
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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 ...
0
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1answer
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 ...
3
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1answer
37 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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0answers
69 views

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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1answer
48 views

topic between algebra and geometry [closed]

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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0answers
39 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
31 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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13 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 ...
0
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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 $a . b = \max(a,b)/\min(a,b)$
3
votes
3answers
45 views

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 ...
2
votes
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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34 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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45 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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0answers
29 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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0answers
19 views

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?
3
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2answers
73 views

Elements of order $3$ in $\text{Aut}\left(\mathbb{Z}/91\mathbb{Z}\right)$

It looks like someone has already been here, but the question I have goes farther. To summarize my work, as well as the work in the above post, we know that ...
3
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2answers
86 views

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 ...
2
votes
3answers
52 views

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 ...
0
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1answer
38 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 ...
1
vote
1answer
80 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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0answers
26 views

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: ...
2
votes
2answers
147 views

Group Theory: group under the composition multiplication modulo $p$

Suppose you have a group $G(S,*)$ where $S=\{1,2,\ldots,p-1\}$, $p$ is prime number, and $*$ is equivalent to the multiplication$\mod p$. If $a,b$ belong to $S$, then $ab\pmod{p}$ also belongs to ...
1
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1answer
33 views

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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1answer
48 views

For $n \geq 2$, find an epimorphism from $S_n$ to $\mathbb{Z}_2$

Find an epimorphism from $S_n$ to $\mathbb{Z}_2$; assume $n \ge 2$. $S_n = \{\iota, \sigma_1, . . . \}$ Let $\phi(\sigma) = \Bigg\{ \begin{array}{cc}0: \sigma = \iota\\1: \sigma \neq \iota ...
3
votes
1answer
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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2answers
59 views

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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1answer
41 views

Kernel of homomorphisms of the Baumslag-Solitar group BS(n,n)

I would like to find the kernel of the following homomorphisms and show that thoses kernels have trivial intersection. $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle$ ...
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0answers
27 views

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 ...
3
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2answers
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. ...
2
votes
2answers
32 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 ...
3
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0answers
36 views

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 ...
6
votes
2answers
79 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...
2
votes
1answer
79 views

Proving a certain lemma about subgroups of $A_n$

In proving $A_n$ is simple for $n\neq4$, my teacher established the cases 1, 2, 3 as obvious, then proved the case 5, and proceded by induction on the rest. In the midst of that induction, he stated ...