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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2
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1answer
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Polynomial representation of elliptic curve points (Frobenius Endomorphism)

I'm trying to understand the Schoof algorithm for counting the number of points on elliptic curves in finite fields. I.e. the most basic algorithm to efficiently determine $\#E(F_p)$. For literature, ...
0
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1answer
30 views
0
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22 views

Kernel of group actions on a set $A$

Let a group $G$ act on set $A$. Show that if $b=g\cdot a$ for $a,b\in A$ and $g\in G$ then $G_b=gG_ag^{-1}$. Find the kernel of this action if G acts transitively on A. I dont know how to approach ...
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3answers
1k views

$gHg^{-1}\subset H$ whenever $Ha\not = Hb$ implies $aH\not =bH$

If $H$ be a subgroup of a group $G$ such that $Ha \not=Hb$ implies that $aH\not=bH$.Then how can I show that $gHg^{-1}\subset H$ $\forall$ $g\in G$? I do not think I have made any progress.However, ...
7
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0answers
32 views

Number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$

For $p$ a prime, what is the number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$, $k \ge 1$? I can answer the $k=1$ case. For each element of $SL_6(\mathbb{Z}/p\mathbb{Z})$, there are ...
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0answers
40 views

a problem in homological algebra

For $C$ is abelian group satisfied $pC=0$ with p is a prime number and $G$ is abelian group. prove that $Ext_{Z}(C,G)\cong Hom(C,G/pG)$ I thought about this in 2 hours but couldn't prove it!
2
votes
1answer
34 views

Littlewood Richardson rules for the orthogonal group SO(d)

I have a question related to tensor products of Young diagrams. More precisely, I know the Littlewood Richardson rules for the general linear group GL(d) and would like to know the restriction of ...
3
votes
3answers
241 views

Matrices as generators of free group.

In the introduction section of the paper Triples of $2\times 2$ matrices which generate free groups the authors mentioning some thing... In my words: The matrices $\begin{pmatrix}1 & 0 \\ 2 ...
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0answers
23 views

Soluble group of derived length 4

Is it true that a non abelian soluble group of derived length at most 4 is locally finite? It looks like this is the conclusion in an article, but I couldn't figure out why.
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1answer
26 views

Abelian-by-(finite abelian) [closed]

hope you all doing fine. I have a question. Is it true that a abelian-by-(finite abelian) group is also (finite abelian)-by-abelian? Thanks.
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0answers
17 views

Automorphism of two members as Generator

Let $X=\langle a,b|a^{2^m}=b^{2^n}=1,[a,b]=a^{2^{m-1}}\rangle$, $m,n\ge 2$ If $\alpha \in Aut(X)$ (Automorphism Group of $X$) is defined as \begin{cases} \alpha(a)=a^{2^{m-1}+1}\\ ...
3
votes
1answer
84 views

Finiteness of subgroup $\rightarrow$ Finiteness of the group

Let $G$ be a group and $H$ be its abelian and normal subgroup. If $H$ is finite and maximal, prove that $G$ is finite. What I tried : Assume $H=\{e,h_2,\cdots,h_{n}\}$. As for each $j$, we ...
3
votes
1answer
32 views

Showing that a finite by cyclic group its automorphism group is finite

Let $G$ be a finite by cyclic group.Prove that its automorphism group $\operatorname{Aut}(G)$ is finite. A finite by cyclic group is a group G that has a normal subgroup $N$ such that $N$ is ...
0
votes
1answer
35 views

Intersection of subgroup of finite index with infinite subgroup is infinite [closed]

Let $H\subseteq G$ be a subgroup of $G$ of finite index. Further, $F\subseteq G$ is an infinite subgroup of $G$. Prove or disprove: $H\cap F$ is infinite.
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1answer
49 views

Determine number of elements of order 12 of a group

Let's say we have a commutative group G that's specified by generators and relations. We find that the group G normal form is: $Z_2\times Z_6\times Z_{12}$ and that the elementary form is $Z_2\times ...
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0answers
29 views

What does it really mean to complexify the $10$-dimensional representation of $ \mathfrak{so}(10)$?

A commonly used "trick" in $SO(10)$ Grand Unified Theories is to use a "complex" instead of a "real" $10$-dimensional representation for the Higgs fields. My problem is understanding what this ...
1
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1answer
402 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
7
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3answers
64 views

Is there any element of order $51$ in the group $U(103)$

Does there exist an element of order $51$ in the multiplicative group $U(103)$ ? Now if the element exist say $x$ then it satisfies the equation $$x^{51}\equiv 1\pmod {103}$$ . ...
1
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1answer
43 views

If $|xH|$ has order $n$, then there is an element $y$ with $|y|=n$ and $xH=yH$

Let $G$ be a group, and let $H$ be a normal subgroup with $|H|=m$. Suppose $n$ and $m$ are relatively prime. If $|xH|$ has order $n$, we wish to find an element $y$ with $|y|=n$ and $xH=yH$. It is ...
0
votes
1answer
16 views

Is there any way I can show $\theta(x)\theta^2(x)=x^{p-1}$, for $\theta\in \operatorname{Aut}(C_p)$ having order $3$

So I am not sure if I am being dim but I am stuck on what seems like a straightforward group theory argument. Suppose we have some $\theta\in \operatorname{Aut}(C_p)\cong C_{p-1}$ such that ...
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0answers
32 views

Show that the homomorph image of an abelian group is abelian

Since $G$ is abelian, we have that: $$ab = ba \implies \phi(ab) = \phi(ba) \implies \phi(a)\phi(b) = \phi(b)\phi(a)$$ Am I rigth?
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3answers
37 views

Axioms for Group Action

Reading the Wikipedia article on group action, I am wondering, why are the axioms stipulating that a group action obey both "compatibility" and "identity"? If a group action is merely a group ...
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0answers
12 views

Complexification of a Lie algebra representation in terms of weights?

EDIT: I found in this book the sentence: The weight system of a real representation of $G$ is defined to be the weight system of its complexification I think if someone can explain what this ...
3
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0answers
167 views

Inverse-closed subsets generating simple groups in finitely many steps

Let $G$ be an infinite simple group. Suppose there is a subset $X$ of $G$ with $$ X^g = gXg^{-1}=X\qquad (g \in G) $$ which is closed under taking inverses and which even generates $G$ in finitely ...
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26 views

For any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ?

Is it true that for any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And is it true that for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ? ( One problem I'm ...
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0answers
37 views

Why is a weight automatically a complex weight?

EDIT: I think the partial answer to my question is that in order to talk about weights we always need a complex Lie algebra. If the Lie algebra is real, we use the complexification, This is necessary, ...
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0answers
38 views

Where to the degrees of freeedom go when a complex representation becomes a real representation of a subalgebra?

As an example consider the complex $16$-dimensional representation of $\mathfrak{so}(10)$. When $\mathfrak{so}(10)$ is reduced to the subalgebra $\mathfrak{so}(9)$, the complex $16$-dimensional ...
3
votes
1answer
70 views

projective geometry and projective space

Let $V$ is a vectorspace over field $F_q$, we denote the set of all subspaces of $V$ by $\mathcal{P}(V)$. I saw some referencess they mentioned $\mathcal{P}(V)$ as a projective space and some ...
1
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1answer
41 views

p-element centralizing a Sylow p-subgroup

Let $G$ be a finite group, $P$ a Sylow $p$-subgroup for a prime $p$ and $g$ a $p$-element with $gxg^{-1} = x$ for all $x \in P$. Then $g \in Z(P)$. Is this true? How can i prove that $g \in P$ ...
0
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1answer
24 views

Meaning of $^sB$, s an element, B a subgroup

Let $G = SL_2(\mathbb{F}_q)$, $B$ the subgroup of all upper triangular matrices, $s = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. What does $^sB$ mean? I read it from page 4 of C. ...
3
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2answers
146 views

Is there an isomorphism of additive groups between $\mathbb{Q/Z}$ and $\mathbb{Q}$?

I know that I have to study the order of every element in $\mathbb{Q/Z}$. But what do I do? I've been struggling of what to do for this question
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2answers
66 views

If $\phi$ is an isomorphism, $\phi(g)^n = 1 \iff g^n = 1$. Doesn't this hold for homomorphisms too?

I need to prove that for an isomorphism $\phi$, the following is true: $$\phi(g)^n = 1 \iff g^n = 1.$$ We know that $$g^n = 1 \implies g\cdot g \cdots g = 1\implies \phi(g\cdot g \cdots g) = ...
2
votes
2answers
39 views

Proof that normalizer and center are subgroups

I've seen this proof for the center of a group $G$: $$C = \{x\in G:xg = gx \ \ \ \forall g \in G\}$$ So, the center is the set of all elements that commute with every $g$ of $G$. This subset of $G$ ...
9
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1answer
59 views

An element of $GL_n(\mathbb F_p)$ cannot have order $p^2$ if $n < p$

I'm preparing for my graduate program's entrance exams, and I came across this problem when studying. Our study group came up with a solution, but I wanted to ask if it was actually correct, since ...
3
votes
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
votes
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 ...
2
votes
2answers
48 views

Determining the minimum dimension required for embedding a finite group

Consider the groups $S_3$ and $S_4$ which are the symmetric groups on 3 and 4 elements respectively. We note that $S_3$ can be realized geometrically as the set of all rotations and reflections of a ...
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3answers
33 views

Show that if $A\cup B = A\vee B$ for subgroups $A$ and $B$, then $A\subseteq B$ or $B\subseteq A$

I have that $$A\cup B = A \vee B$$ My book defines $A \vee B$ as being: $$\cap\{T: \text{T is a subgroup of $G$ and $A\cup B \subseteq T$}\}$$ So, if I take the intersection of all subgroups ...
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votes
1answer
26 views

transitive action on finite abelian subgroups [closed]

Let $G$ be a group and $K$ a finite subgroup of $G$. Let $H$ be some subgroup of the normalizer of $K$ in $G$, and assume the action of $H$ on $K$ by conjugation is transitive on elements of $G$ of ...
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1answer
28 views

Verifying if these Cayley tables are from groups

For the first table I noticed that $ab = c \implies abb = cb \implies a = cb$ but in the table, $cb = d$, so this can't be a group For the second table, we have: $ab = c \implies (aa)b = ac ...
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1answer
20 views

$S$ is a non empty set and there are $a$ and $b$ for $c$ and $d$ such that $a\cdot c = d$ and $c\cdot b = d$, prove it is a group

An associative operation $\cdot \ $was defined in $S$ such that $\cdot \ $is associative. Also, for all the pairs $c$ and $d$, there are elements $a$ and $b$ such that: $$a\cdot c = d, \ \ \ \ c\cdot ...
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0answers
46 views

How is the delooping of a groupoid constructed explicitly?

Let $G$ be a group, nlab's "delooping" page says that $G$ can be considered as a discrete groupoid in the $(\infty,1)$-topos $\infty$Grpd of $\infty$-grupoids, the delooping of $BG$ is then the ...
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1answer
34 views

Two sub-groups of order 3 and 5, prove that the group of order 15 is cylic. [closed]

So all I have is that G is a group of order 15, and there are 2 unique sub-groups, which order is 3 and 5 (I mean there only one sub-group of each kind) and I need to prove that G is cyclic. Dont see ...
3
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0answers
28 views

Proving a version of the Kronecker's Theorem

I am trying to prove the following version of the Kronecker's Theorem: Suppose $k$ is a positive integer and $\{1, \theta_0, \dots, \theta_{k-1}\}$ is linearly independent over $\mathbb Q$. Then ...
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0answers
39 views

$G$ of finite order $2p$ ($p$ is prime). Prove that $G$ abelian. [duplicate]

I have a group $G$ of order $2p$, where $p>2$ and prime. The additional thing that I also know, that $\exists a\in Z(G)\mid O(a)=2$. I need to prove that G is abelian. But first, before that, ...
3
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0answers
43 views

Standard notation for indices in group theory?

I've seen three notations for indices in group theory, namely $(G:H)$, $[G:H]$ and $|G:H|$. Is there any of these notations that is standard?
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1answer
40 views

About a proof regarding a property of groups of order $pq$ where $p$ and $q$ are primes

I'm studying right now Automorphisms in Dummit & Foote's Abstract Algebra (Section 4.4). In pages 135-136, the following example is given: and here's Proposition 16 muntionned in the ...
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0answers
23 views

Finitely generated abelian groups and finite index subgroups

I want a proof or reference for the following fact: "Let $G$ be a finitely generated abelian group and let $\phi:G\to G$ be an injective homomorphism. Then the index $[G:\phi(G)]$ is finite." I ...
1
vote
1answer
49 views

Discontinuous $ f : \mathbb R^2 \to \mathbb R$ with unusual topology on $ \mathbb R$

With the usual topology on the reals $\mathbb R$ , let $D$ be the family of dense open sets and let $T=D \cup \{ \phi \}$. Let $S$ be the set $R$ with the topology $T$ on it. Show that the function ...
2
votes
1answer
34 views

Presentation of Abelianization of a group

Say $G$ is a finite group with presentation $\langle S | R \rangle$ and let $C$ be the commutator subgroup of $G$. Then $\langle S | R \cup \{ sts^{-1}t^{-1} \} \rangle$ is a presentation of $G/C$. ...