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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2answers
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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.
7
votes
0answers
153 views

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 ...
3
votes
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
20 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
votes
0answers
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 ...
1
vote
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 ...
1
vote
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 $M:G_1\times G_2\to G_3$ ...
0
votes
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 $ab \in A \cap B$?
1
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2answers
34 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
votes
2answers
59 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 : ...
1
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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, ...
3
votes
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 ...
3
votes
1answer
42 views

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 ...
0
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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?
4
votes
1answer
71 views

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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0answers
42 views

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 ...
8
votes
2answers
172 views

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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0answers
37 views
+50

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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0answers
30 views

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 ...
2
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1answer
27 views

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 $K$: $$\operatorname{Tran}_G(H,K)=\{g\in G\mid gHg^{-1}\le K\}$$ ...
2
votes
1answer
39 views

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 ...
2
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2answers
34 views

$P$ 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 ...
0
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2answers
30 views

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 ...
0
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1answer
39 views

The trivial homomorphism [closed]

Please help me I need to prove that for any positive integer n, the only homomorphism between the cyclic group of order n and the complex number under addition is the trivial homomorphism
0
votes
1answer
54 views

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 $[G:K]$ ?. ...
3
votes
1answer
44 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 ...
0
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0answers
34 views

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 ...
4
votes
1answer
53 views

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 ...
0
votes
2answers
31 views

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)?
2
votes
1answer
45 views

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 ...
0
votes
1answer
38 views

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 ...
2
votes
1answer
57 views

Can the possible groups by determined by hand?

Suppose, $G$ is a group of order $12$ containing an element $a$ with order $4$. Can I show the following facts by hand ? The group is either cyclic or isomorphic to the group $C3:C4$ $a^2$ is the ...
1
vote
1answer
25 views

An inequality for the minimal number of generators of a finite group II

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating exactly the left regular representation (with ...
4
votes
2answers
19 views

Conjugate elements of $SO(3)$ group

Composition of two rotations in 3d space yields another rotation $$R_1 R_2 = R_3, $$ and I can understand this by help of some figures in my book. So, the rotations in 3d space forms group. Then ...
2
votes
1answer
55 views

Quotient $G \to G/N$ induces quotient $H \to H/N$ by restriction?

Let $G$ be a linear algebraic group over an algebraically closed field $k$. Consider closed subgroups $N \subseteq H \subseteq G$ such that $N$ is a normal subgroup of $G$. Then restricting the ...
2
votes
1answer
28 views

If $\alpha,\beta \in S_{n}$, and $\alpha\beta = \beta\alpha$, then $\beta$ permutes those elements left fixed by $\alpha$.

Here is my solution. Let ${a_1,...,a_k}$ represent all the integers that are permuted by $\alpha$, and let ${a_{k+1},...,a_{k+j}}$, where $k+j \leq n$, be all the elements that are left fixed by ...
1
vote
1answer
24 views

Representatons of dimension $1$ on $D_4$

Prove that there are $4$ distinct representations on $D_4$ with dimension $1$ (where the field is $\mathbb{C}$). We have just started learning representations. Getting this question, what ...
0
votes
1answer
27 views

Problem 4 of Section G of Chapter 13 of Pinter's Book of Abstract Algebra

First, some background info/context: Let $G$ be any group of order $10$. Then, by Cauchy's Theorem, there are elements $a, b \in G$ such that $\text{ord}(a)=2$ and $\text{ord}(b)=5$. Since ...
1
vote
0answers
25 views

Baer-Specker group versus free abelian group generated by an uncountable set [duplicate]

I just learned on Wikipedia that the Baer-Specker group, that is, the group of all integer sequences, is not free abelian. I'm hoping I could be helped to understand why this is true by someone ...
1
vote
1answer
47 views

Prove that (possibly) infinite group of all invertible maps of X to itself is not Abelian.

I have this question on my assignment and I this fact seems trivial to me, but I can not come up with a rigorous proof. I thought to go by contradiction: Assume such a group $G$ is Abelian -> ...
2
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1answer
25 views

Looking for some insight or explanation in Normal group and similar matrix

I need someone can give me some insight in Normal group and Matrix Similarity. Normal subgroup $g \in G \text{ and } N \lt G$ $gNg^{-1} = N$ Matrix Similarity $B = PAP^{-1}$ It seems to me ...
4
votes
2answers
81 views

Why isn't a set like $\{0,1,3,6,8\}$ a subgroup of $\mathbb{Z}_9$?

Why are there only 3 subgroups of $\mathbb{Z}_9$? What about $\{0,1,3,6,8\}$? There is an identity and inverse for each element in that subset.
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0answers
45 views

Group Action: Group $(\mathbb{Z}, +)$ acting on $\mathbb{R}$

In my group theory notes I have the following: The Group $(\mathbb{Z}, +)$ acts on $\mathbb{R}$ as follows: $m\in \mathbb{Z}$ and $r\in\mathbb{R}$: $m.x \to (-1)^mr$ in this notation $m.x$ ...
4
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1answer
28 views

Group theory commutator and solvable groups

let G be a group such that it contains 2 members $a, b \in G$ that statisfy: $a = p^{-1} b p$ where $p \in G$ $a = q^{-1} [a,b]q $ where $q \in G$ $a,b,[a,b]\neq e$ where $[a,b]$ is the commutator ...
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0answers
35 views

Can the number of order-sequences of a group of order $n$ be determined efficiently?

Let $n\ge 1$ and $G$ a group of order $n$. Determine the order of every element of $G$ and list the orders increasing. Can the number of possible order-sequences be determined efficiently ? For ...
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1answer
46 views

Proving isomorphism [duplicate]

I want to prove at any group of order $4$ is isomorphic to either $\Bbb Z_4$ or $\Bbb Z^*_8$. I know that these two groups are not isomorphic to each other because they have different order, but I ...
1
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1answer
36 views

representation of symmetric group

Please help me for the following statement. I do not understand what does it mean centralizer of a representation. The imprimitive transitive representation of symmetric group $S_n$ of degree 2n has ...
0
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0answers
20 views

Prove that the order of $GL_{2}(\mathbb{{F}_{p}})$ is $p^{4}-p^{3}-p^{2}+p$. [duplicate]

Let $p$ be a prime. Prove that the order of $GL_{2}(\mathbb{{F}_{p}})$ is $p^{4}-p^{3}-p^{2}+p$. I know that the matrix $2\times 2$ is given by this formula ...
0
votes
1answer
27 views

Abelianization of $\mathbb{Z}_2*\mathbb{Z}_3$

Intuitively it has to be $$\text{Ab}(\mathbb{Z}_2*\mathbb{Z}_3)=\mathbb{Z}_2\times\mathbb{Z}_3$$ here is my approach on how to prove it $$\mathbb{Z}_2=P(a\mid a^2),\mathbb{Z}_3=P(b\mid b^3)\Rightarrow ...
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votes
0answers
27 views

General linear group acting on vector space

I have this question really stuck on it Let G denote general linear two by two matrices over field Fp for a prime p acting on the vector space of two column vectors over Fp i cannot find orbits or ...