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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What's the name for the property for which $x + x = 0 \Longleftrightarrow x = 0$?

I have a set $\mathbb{S}$ for which I have defined an operation: addition ($+ : \mathbb{S} \times \mathbb{S} \rightarrow \mathbb{S}$). The structure $(\mathbb{S}, +)$ is a group. I have shown that if ...
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2answers
49 views

If $G \cong \mathbb Z/3\mathbb Z$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$

Let $G = \{1,g,g^2\}$ be the cyclic group of order three. Consider the group ring $\mathbb R[G]$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$ with the isomorphism $$ \varphi(1) = (1,0), ...
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34 views

$G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$?

Let $G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$ ? ( I know that any subgroup of index smallest prime dividing ...
2
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0answers
22 views

Topological/geometric interpretation of conjugacy separability

We say that a group $G$ is residually finite if for every non-trivial $g \in G$ there is a a finite group $Q$ and a surjective homomorphism $\phi \colon G \to Q$ such that $\phi(g) \neq_Q 1$. This ...
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2answers
57 views

Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ?

Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ? ( I know that there 'is' a 'surjection' , but I don't know whether any surjective homomrophism from $\mathbb R$ ...
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0answers
31 views

Quotient of $S_4$ by a normal subgroup

Is this true? > Quotient of $S_4$ by a normal subgroup of order 4 is Abelian ? As the group is of order 4!/4=6, so it is either $S_3$ or $\mathbb Z_6$. how to decide? please help.
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29 views

Simple non-abelian subgroup

How can I show that If H is a simple and non-abelian subgroup of group G then $ H< [G,G] $.THank you for all your answers.
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20 views

Image of Hall subgroup is a Hall subgroup

Let $\pi$ be any set of prime numbers. $H \subset G$ is a Hall $\pi$-subgroup if no pime dividing the index $|G:H|$ lies in $\pi$ and every prime divisor of $|H|$ lies in $\pi$. Let $\varphi: G ...
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0answers
24 views

freeness of vector spaces and abelian groups

This question is continuation of my previous question Extension of vector spaces and abelian groups Given a diagram of linear transformations of $K$ vector spaces $$B\xrightarrow{\epsilon} ...
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1answer
29 views

Prove that, considering $\mathbb{R}$ as an additive group, we have $SO_2(\mathbb{R})\cong \mathbb{R}/2\pi\mathbb{R}$ .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. Prove that, considering $\mathbb{R}$ as an additive group, we have $SO_2(\mathbb{R})\cong ...
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1answer
58 views

Alternating group on infinite sets

It is well known that the only normal subgroup of $S_n$ is $A_n$ when $n\geqslant 5$, and that $A_n$ is also simple. Furthermore, $A_{\infty}$, the even permutations on $\mathbb{N}$, is also simple. ...
0
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0answers
16 views

Why is $(A \otimes_\mathbb{Z} V_p )\bigcap R(G) = V_p$

I am reading the proof of brauers theorem in Serres book Linear Representations of finite groups and I have trouble understanding Lemma 5(page 75). Let $G$ be a group of order $g$. First let $g=p^nl$ ...
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1answer
58 views

Where does the group $\mathbb Z/(a)\oplus \mathbb Z/(a^2)\oplus \cdots $ arise?

Let $a>1$ be an integer, and consider the infinite abelian group $$ V_a=\bigoplus_{j=1}^{\infty}\mathbb Z/{a^j\mathbb Z}. $$ Can anyone provide references to places where this (or related) groups ...
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1answer
45 views

Is the definition given by the GAP-manual equivalent to the one given in the site? [on hold]

Here http://groupprops.subwiki.org/wiki/Normalizer_of_a_subset_of_a_group the normalizer of a subset of a group is defined. GAP gives the following description of the Function IsNormal : 39.3.6 ...
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21 views

A normal subgroup of a finite $p$-group is in the center

I am trying to prove the following (it is an exercise from Rotman): Let $G$ be a finite $p$-group and $N$ its normal subgroup of order $p$ ($p$ is a prime number). Then $N$ is a subgroup of the ...
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0answers
22 views

oligomorphic subgroups of $S_\infty$

Is it true that every oligomorphic subgroup of $S_\infty$ is not abelian? A subgroup of $S_\infty$ is said oligomorphic if its action on $\mathbb N^n$ has only finitely many orbits for each ...
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1answer
17 views

How can we prove that the group order is an upper bound both for the number and the dimensionalities of the irreducible representations?

For example, in $S_{3}$ there is 6 number of members for the group while there are 3 different irreducible representation with dimensinalities of 1, 1 and 2 in which gives: $$\sum_{\mu}{n_{\mu}^2} = ...
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0answers
34 views

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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30 views

$G_\delta$ 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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1answer
36 views

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
37 views

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
33 views

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 ...
2
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0answers
18 views

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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0answers
27 views

Proof that $\exp(a) \cdot r \cdot \exp(-a) =\ exp(ad_a) \cdot r$ [on hold]

Let a nilpotent element of associative algebra over field with zero characteristic and $ad_a: b \to [a, b] = ab -ba,$ $b \in R $. Proof that $\exp(a) r \exp(-a) = \exp(ad_a)r (r \in R) $
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33 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 $\mathbb{Z}/p$ [on hold]

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
33 views

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

Show by giving an an example that Maschke's theorem does not hold for all infinite groups.
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1answer
56 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 ...
6
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1answer
47 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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2answers
50 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
20 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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0answers
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
47 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 ...
0
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1answer
40 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 ...
4
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0answers
46 views

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
33 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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2answers
32 views

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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+200

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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3answers
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Textbooks for Groups & Rings [closed]

Please I need suggestions on the best textbooks to help me comprehend this Groups and Rings and relate it with the rudimentary aspect of Set Theory
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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.
2
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1answer
19 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 ...
3
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0answers
47 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
48 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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1answer
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
65 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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2answers
33 views

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 ...
2
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2answers
57 views

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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0answers
25 views

$S_n$ 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, ...