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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about a finite by nilpotent group

I was reading the proof of the following lemma let $G=XM$ xhere $M$ is a normal divisible abelian subgroup and $X$ is a torsion subgroup of $G$.If $G$ is a finite by nilpotent group then ...
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$G_\delta$ subgroups of a Polish Group

Let $X$ be a Polish Group. It's known that every its Polish subgroup is a $G_\delta$. Pick one of them, say $V$. Is it true that $V$ is the intersection of open subgroups? Thank you
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Polish subgroups of $S_\infty$

Let $S_\infty$ considered as Polish Group. Prove that every Polish subgroup of $S_\infty$ has the following form: $\overline{{\left \langle X \right \rangle}}$, where $X$ is a countable subset of ...
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1answer
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If $G = G_{\alpha} \rtimes R$ and $R$ is non-abelian of order $27$ and exponent $3$, then $G_{\alpha} \cong C_2\times C_2, D_3$ or $D_4$

Suppose that $G$ is a transitive permutation group and let $R$ be a regular normal subgroup which is isomorphic to the non-abelian group of order $27$ and exponent $3$. Then $G = R \rtimes ...
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2answers
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Extension of vector spaces and abelian groups

While reading about modules from Hilton & Stammbach's Homological algebra, I saw the following statement : $\Lambda$ is a ring. $\Lambda$ modules are generalizations of vector spaces and ...
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Factorization of permutations into two factors with fixed number of cycles, plus a placement condition

I have asked this question in MathOverflow, but it received no answers, so I am posting it here. In my recent work I have been led to consider the following type of permutation factorizations. Let ...
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34 views

Origin of the name “Cauchy sequence” in abstract algebra

I know the origin in analysis as it is derived from our good old chap Cauchy himself. However I have read about Cauchy sequences in algebra, I'll use groups for this one, let $G$ be a group and ...
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1answer
44 views

The number of elements in the special linear group over the finite field $\mathbb{Z}/p$ [closed]

I have $SL_{2}\{\mathbb{Z}/p\}$ for $p$ prime and $\mathbb{Z}$ integers. How do I show that this is a subgroup of $GL_{2}\{\mathbb{Z}/p\}$ and find the number of elements in $SL_{2}\{\mathbb{Z}/p\}$?
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1answer
38 views

example of infinite group that maschke's theorem is not hold [closed]

Show by giving an an example that Maschke's theorem does not hold for all infinite groups.
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57 views

Construction of a Specific Non-commutative and Infinite Group

I am struggling with the following problem: Find a group $G$ such that whenever $m, n, k \geq 2$ are natural numbers, then there exist $a, b \in G$ such that the order of $a$ is $m$, order of $b$ is ...
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52 views

An Analogue of Chinese Remainder Theorem for Groups

I am trying to prove the following analogue of Chinese remainder theorem for groups: Let $G$ be group and let $H_1, \dots, H_n$ be its normal subgroups such that their indices $[G : H_1], \dots, [G : ...
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51 views

Prove that a group G is finitely generated if and only if there is a surjective homomorphism $F(\{1,..,n \}) \to G$

This if from Aluffi's Algebra: Chapter 0. There is an another definition of subgroup generated by a subset. Here it is: Let $G$ be a group and $A$ its subset. By universal property of a free ...
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1answer
21 views

Understanding a terminology in a special type of group

I am trying to understand the following terminologies, and the resulting group (found in this link). In the original reference also, I didn't find the meaning of the terminology I am looking. It is ...
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1answer
35 views

If $[G:H\cap K]= [G:H][G:K]$ then $G=HK$.

Let $G$ a finite group and $H,K$ subgroups of $G$. Show that if $[G:H\cap K]= [G:H][G:K]$ then $G=HK$. I proved that $G=HK$ implies $[G:H\cap K]= [G:H][G:K]$ but not the other direction. Thanks for ...
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24 views

$G/\lambda_2$ is nilpotent?

Let $G=\lambda_0 \geq \lambda_1 \geq \lambda_2 \ldots$ where $\lambda_i=[\lambda_{i-1} , G]$ be the lower central series. Then is it true that $G/\lambda_2$ is nilpotent? If true how can I prove this. ...
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1answer
51 views

What are the units in $\mathbb{Z}/n\mathbb{Z}$ in general?

What are the units in $\mathbb{Z}/n\mathbb{Z}$ ($n$ is any positive integer) in general? I figured it should a group under multiplication mod $n$, but was wondering if there is any more specific way ...
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1answer
42 views

Can anyone explain how primitive roots work?

Right now I'm studying out of Audrey Terras' book Fourier Analysis on Finite Groups and Applications and we're on the section where we're talking about $(\mathbb{Z}/n\mathbb{Z})^*$ and when this group ...
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Prove that there exist a sylow subgroup of $G$ which is fixed by $\alpha$.

Let $G$ be a finite group and $\alpha\in Aut(G)$ such that $\alpha$ restricted to any sylow subgroup of $G$ equals the restriction of some inner automorphism of $G$ and $(o(\alpha),o(G))=1$, then is ...
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1answer
34 views

On Simple Algebraic Groups

I was skimming a paper and got stuck in the middle. As you see in the underlined parts, the authors first assumed that $\mathcal{G}$ is a simple algebraic group. Then $\mathcal{G}$ is defined to be ...
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1answer
26 views

Prove that locally finite nilpotent group is direct product of its normal maximal $p$- subgroups

Definition- A group $G$ is said to be locally finite if every finite subset of $G$ generates a finite subgroup. Now I have to prove the following proposition. Proposition- Let $G$ be a ...
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Theorem on the interpretation of the ring $Z_n$ [closed]

$(m, n)=1 \Rightarrow Z_{mn} \cong Z_m$ x $Z_n$ Can anyone please help me with proof for this theorem? Thanks in advance.
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Inverse (finite group) isomorphism of a certain form exists

I have been working something in group theory for a long time and I have everything worked out but this one problem. I have reduced that problem to a conjecture. It takes some work to set it up, but I ...
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1answer
47 views

If the number of elements in a group $G$ of order $289$ is $n\geq 273$ then $G$ is not cyclic.

Let $G$ a finite group and $n$ the number of elements in $G$ of order $289$. Show that if $n\geq 273$ then $G$ is not cyclic.
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1answer
25 views

Help showing $H_n=\sum_{h \in H}h \in F[H]$ is a simple submodule

Given $F$ a field and $H$ a finite group and $H_n=\sum_{h \in H}h \in F[H]$, I'm trying to show that the left ideal $(H_n)\subset F[H]$ is a simple submodule. Now I'm not really sure what the ideal ...
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48 views

What dihedral subgroups occur in the affine general linear group $AGL(2,3)$

I am interested in the subgroup structure of the affine general linear group $AGL(2,3)$, in particular I want to know if they could have dihedral subgroups other then $D_3$ and $D_4$, i.e. the ones of ...
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1answer
52 views

Is $SL_n(\mathbb{R})$ actually simple?

It's probably not hard to prove that $\frak{sl}_n\mathbb{R}$ is simple, so that $SL_n\mathbb{R}$ has no nontrivial connected normal subgroups. But do there exist discrete normal subgroups of ...
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25 views

Need information on the following multi-group homomorphic structure

For the sake of simplicity I will describe the problem with a three group structure. Suppose there are three groups $G_1$, $G_2$ and $G_3$. Suppose also there is a binary map $M:G_1\times G_2\to G_3$ ...
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1answer
75 views

Intersection of two normal subgroups of a group

Let G be a group, and let A,B be normal subgroups of G. If $a \in A$ and $b \in B$, does this mean that $ab \in A \cap B$?
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Is there a difference between modulo groups with and without asterisks ($\mathbb{Z}_{38}$ vs $\mathbb{Z}_{38}^*$)?

I know modulo group $\mathbb{Z}_{38}$ but I saw it with a star in some question: $\mathbb{Z}_{38}^*$. Is it is the same as $\mathbb{Z}_{38}$ or a different group? If it refers to the same group does ...
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Prove that $H\cap C$ is non-empty for every conjugacy class of $G$

Let $G$ be a finite group and $\phi\in Aut(G)$ such that $\phi^q=id$ where $q$ is prime and does not divide $|G|$. Moreover $\phi$ preserves conjugacy classes of $G$. Then consider $H=\{g\in G : ...
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$S_n$ is isomorphic the permutations of the identity matrix?

Prove that the set of permutations $S_n$ is isomorphic to the group of invertible square matrices of order $n$ where each row has $n-1$ zeros and $1$ in one place. This is very intuitive to me, ...
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1answer
55 views

Why can quotient groups only be defined for subgroups?

I know that for the operation on cosets to be well-defined one requires normality. But why is it a requirement (with $G / N$) that $N$ be a subgroup or even a subset of $G$? Surely all that is ...
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Can I use GAP to show block structures in a multiplication group clearer $\ $?

The usual output of GAP for the multiplication table of the group $S3$ is ...
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1answer
37 views

Permutation representations of symmetric group of small order

Thanks for any comments or help. What is the list of faithful permutation representations of $S_k$ of degree at most $n=2k$ for $k=3,4,5,6$? Is it possible to find its centralizer in each case?
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Subgroups of finite index of $\mathbb Z/2\mathbb Z\times \mathbb Z$

Let $H$ be a subgroup of index $n$ in $(\mathbb Z/2\mathbb Z)\times \mathbb Z.$ Is there finitely many subgroups of finite index of $(\mathbb Z/2\mathbb Z)\times \mathbb Z$ ? If yes, can we ...
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Index of free group of rank $k$ in free group of rank $n$

I thought about the following question What is the index of the free group of rank $k$, denoted $F_k$, in the free group of rank $n$, denoted $F_n$? Let's say for the moment, $k < n$, and ...
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How are simple groups the building blocks?

I know a bit about simple groups. A finite Abelian group is a (direct) product of finite cyclic groups. The simple finite Abelian groups are exactly $\mathbb{Z}_p$ for $p$ a prime. And so, I ...
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If $M < M < G$ with certain conditions and a special subgroup $U < G$, then we can choose $|G / M| = p$ and $p \nmid |M|$.

Let $G$ be a finite group and $U \le G$ a subgroup of odd order. Assume that $N_G(U) = TU$ with $T = \langle t \rangle$ for some involution $t \notin U$. Also assume $U^g \ne U$ implies $U^g \cap U = ...
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Exercice about the group of unit $(\mathbb Z/21\mathbb Z)^\times $.

Let $(\mathbb Z/21\mathbb Z)^\times$ the group of units of $\mathbb Z/21\mathbb Z$. 1) How many element in $(\mathbb Z/21\mathbb Z)^\times$ ? 2) Is it isomorphic to an abelian group ? 3) Is it ...
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1answer
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A description of the transporter $\operatorname{Tran}_G(H,K)$ for subgroups $H\le K$

Let $H$ and $K$ be subgroups of a group $G$. The transporter of $H$ into $K$ is the set of all $g\in G$ that conjugate $H$ into $K$: $$\operatorname{Tran}_G(H,K)=\{g\in G\mid gHg^{-1}\le K\}$$ ...
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1answer
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Finding the homomorphisms $S_3 \to \Bbb Z / 6 \Bbb Z$

I have to find explicitly (i.e. as operated on the element of the domain) the homomorphisms (of groups) from the symmetric group $S_3$ to $\Bbb Z / 6 \Bbb Z$. Do I study the possible kernels of the ...
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$P$ is a Sylow subgroup of $G$ then $P$ is normal in $G$.

Let $G$ be a finite group with subgroups $H$ and $P$ and if $H$ is normal in $G$ and $P$ is normal in $H$ and $P$ is a Sylow subgroup of $G$ then $P$ is normal in $G$. Is the statement true, I heard ...
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Inverse of the element in the multiplicative group

I need to show that if $k\in \Bbb Z^*_n$ has an inverse with respect to the multiplication modulo, $n$, then $k,n$ are coprime. Can anyone give me a hint how to use the fact that $k$ has an inverse ...
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Index is multiplicative

Let $G$ be a group and satisfies minimal condition on subnormal subgroups. Further let $H, K\trianglelefteq G$, such that $H\leq K$ and $[G:H]$ is finte, then can we say something about $[G:K]$ ?. ...
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45 views

Is it enough to determine if two finite groups are isomorphic if we can map the generators?

Heading: A General Inquiry about Finite Isomorphic Groups and Their Generators If I am given two finite groups whose generators I know, is it enough to show that by mapping the generators ...
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a representation of permutation groups

The smallest degree of faithful permutation representations of $S_k$ for $k\geq 6$ have degrees $k$ (natural action), $2k$ (imprimitive action on cosets of $A_{k−1} )$, and $k(k − 1)/2$ (action on ...
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Automorphisms of a group and subgroup

Is there a finite $p$-group $G$ such that $G$ has less number of automorphisms than some subgroup: $$|\mbox{Aut}(G)|<|\mbox{Aut}(H)| \mbox{ for some }H\leq G.$$ If there is such a group, then can ...
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Is there a cyclic group such that is isomorphic to Z∗16? [closed]

How do I find an additive group modulo $n$ $Z_n$ which is isomorphic to $Z^*_{16}$ (group with multiplication modulo 16)?
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1answer
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Automorphism group from non-abelian simple group

Denote $\varphi_G: G \rightarrow Aut G, a \mapsto (x\mapsto axa^{-1})$. I have shown that if $\varphi_G$ is injective, then we have $\varphi_{Aut G}$ is injective. I am now asked to prove that for ...
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Is the dihedral group Dn linearly primitive for n>2?

Let $D_n$ be the Dihedral group (of order $2n$). For $p>2$ a prime number, $\mathbb{Z}/2$ is a core-free maximal subgroup of $D_p$, then $D_p$ is a primitive permutation group, and so linearly ...