Tagged Questions

The study of symmetry: groups, subgroups, homomorphisms, group actions.

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24 views

Group of Homomorphism

Given a group homomorphism $\psi: A_8 → S_9$ for which there exists $\sigma \in A_8$ with $\psi(\sigma)=(12)$, prove that $\psi$ is injective Things that I know: If $\psi:G\to H$ is a group ...
1
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1answer
27 views

Building intuition in group theory

I'm finding it hard to translate abstract results of group theory into something that intuitively makes sense. Putting this into a concrete example: if $f:G\to H$, $Im(G)$, is a subgroup of $H$? Is ...
1
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1answer
17 views

Fast way of finding a homomorphism

Show that the dihedral group $D_{12}$ is isomorphic to the direct product $D_6\times C_2$ . I can do it by just checking that each element in one has an equivalent element in the other, but is there a ...
0
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0answers
37 views

Orbit-Stabilizer Theorem

I've received this question for a homework assignment, and I'm getting stuck at the last part :) So I've proved that $H = \{f_m,0 \vert m \in \mathbb{R}^*\}$ is a non-normal subgroup of $A1$ ( the ...
2
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0answers
8 views

Prove that the Hirsch rank of a group is unique

A group $G$ is called polycyclic if there exists a subnormal series $G = G_0 \unlhd G_1 \unlhd \dots \unlhd G_n = \{e\}$ whose factors are cyclic. Prove that arbitrary two polycyclic series of $G$ ...
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2answers
17 views

Find two distinct permutations in $S(4)$ which commute, are not disjoint and neither is the identity and they are not inverses of each other.

Find two distinct permutations in $S(4)$ which commute, are not disjoint and neither is the identity and they are not inverses of each other. I haven't been able to figure out which permutations will ...
0
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0answers
21 views

Quotients of linear algebraic groups in different categories?

I have a question on quotients of linear algebraic groups. Let $G$ be a linear algebraic group and $H$ a linear algebraic group acting on $G$ as an algebraic group. I would like to know what the ...
0
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0answers
20 views

give an example of finitely generated not abelian group which has subgroup of infinite index and not finitely generated.

we know that if $G$ is finitely generated group and $H$ is subgroup of it which has finite index then $H$ is finitely generated, and we know that if $G$ is abelian and finitely generated then every ...
0
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1answer
27 views

Density of Sylow subgroups

Let G be any group of order n.Also assume p be the largest prime dividing n.Let n(p) be the maximum no of sylow subgroups a group of order n can have.Is it possible to sa anything definite about the ...
0
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2answers
20 views

Prove that $G$ is the internal direct product of $H$ and $K$

Let $G$ be a group of order $20$. If $G$ has a subgroups $H$ and $K$ of orders $4$ and $5$ respectively such that $hk=kh$ for all $h \in H$ and $k \in K$, prove that $G$ is the internal direct product ...
0
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1answer
17 views

CHECK: Let $G=\frac{(\mathbb{Z},+)/12\mathbb{Z}}{3\mathbb{Z}/12\mathbb{Z}}$. How many elements are there in $G$?

Let $G=\frac{(\mathbb{Z},+)/12\mathbb{Z}}{3\mathbb{Z}/12\mathbb{Z}}$. How many elements are there in $G$? Write them down explicitly. \end{prob} By the 2nd Isomorphism Theorem, ...
1
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0answers
36 views

Order of a Group with certain elements of composite order [duplicate]

If a group $G$ has just $4$ elements of composite order and their orders are $4$ or $6$, then what can we say about the order of the group? Can we say |$G$| $\leq 16$? Can we say the divisor of the ...
1
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1answer
36 views

Group of order $18$ contains exactly one subgroup of order $9$

I'm trying to prove the following: Proposition: A finite group $G$ of order $18$ has a unique subgroup of order $9$. Here is my attempt: Observe that $18 = 3^2 \times 2$. Let's count the number ...
0
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0answers
28 views

What is explicit form of this kernel?

Let $G$ be a group and $N$ be a normal subgroup of $G$. Let $F$ and $S$ be a free group such that $F/R=G$ and $S/R=N$ for some normal subgroup $R$ of $F$. The map from $N \rtimes G$ to $G$ given by ...
1
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0answers
20 views

What is the structure of the group with at most five elements of composite order?

What is the structure of the group with at most five elements of composite order? For instance can we say any thing about the order of such a group?
1
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3answers
46 views

A question about the proof of a theorem in Representation theory of groups

My Question is about one part of the proof of theorem in the book "A Course in the Theory of Groups" by Derek J.S. Robinson. I highlight the part that my question is about. We know that if $G$ is a ...
0
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0answers
15 views

How can I prove that every group of order 4 is Abelian. [duplicate]

Prove that every group of order 4 is Abelian. I heard the proof is just 3 lines but I don't know how to proceed. I tried proving it by showing it is isomorphic to a group of permutations, but got ...
3
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0answers
27 views

How can i prove that an element of order 5 is a 5 - cycle in S7 group? [on hold]

Please prove that an element of order 5 is a 5 - cycle in any S n group. I am absolutely lost.
3
votes
1answer
50 views

Simple group with Klein four Sylow

If $G$ is a simple group, with a Sylow $2$-subgroup isomorphic to the Klein four group $\mathbb{Z}_2 \times \mathbb{Z}_2$, then I want to show that any two involutions in a given Sylow $2$-subgroup ...
2
votes
2answers
56 views

What is this group explicitly?

Let $G$ be finite group act on a set $X$ transitively. I already proved the set $\{ f : X \to X | f(g*x) = g*f(x) \, \forall x \in X, g \in G \}$ is a group. My question is what is this group ...
2
votes
2answers
23 views

Proving the existence of a homomorphism $\overline f:G/H\rightarrow G'$ such that $\overline f \pi = f.$

I'm working on a problem where I'm given that $G$ is a group, $H$ is a normal subgroup of $G,$ $f:G\rightarrow G'$ is a homomorphism, and $H\subseteq \ker(f).$ I need to show that there exists a ...
2
votes
1answer
90 views

If a finite group acts transitively on a set, does its center also acts transitively?

If $G$ is a finite group acts transitively on a set $X$. Does the center $Z(G)$ also acts on $X$ transitively? I don't see how this statement will be true but I can't come up with a counter example ...
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0answers
54 views

Does a group with $|G| = 33$ have to contain an element of order $11$?

A group with $|G| = 33$ must contain an element of order $11$. Prove or disprove. This is inspired by another MSE question. So we know that there must be an element with order 3. I tried using ...
0
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1answer
21 views

Prove: If $G$ has an element of order $k$ and an element of order $m$, the order of $G$ is a multiple of lcm(k,m).

Prove: If $G$ has an element of order $k$ and an element of order $m$, the order of $G$ is a multiple of lcm(k,m). This is the same as asking to show if $k\mid n$ and $m\mid n$ then $q \mid n$. Where ...
2
votes
1answer
40 views

For finite group $G$ when is $|Aut(G)| < |G|$?

If $G$ is a finite group then we know $|Aut(G)|$ divides $(|G|-1)!$ ; I want to ask , if $G$ is a finite group with more than one element then is it true that $|Aut(G)| < |G|$ ( I know it is ...
0
votes
2answers
29 views

Do the isomorphism's of groups form an equivalence relation on the class of all groups?

An isomorphism is simply a bijective homomorphism. How would one show that isomorphism's are symmetric, reflexive, and transitive?
0
votes
1answer
23 views

Show that if $G$ is cyclic then so is $H$

If group $G$ is isomorphic to group $H$, show that if $G$ is cyclic then so is $H$. An isomorphism is simply a bijective homomorphism. The latter is a function which preserves the group operation. ...
3
votes
1answer
15 views

Finding Sylow p-subgroups

I am trying some examples of finding Sylow p-subgroups in specific groups and looking for the most efficient way to do so. For example, lets say we need to find Sylow-3 subgroups in $A_4$ and $D_6$, ...
3
votes
1answer
26 views

Is it true that $n_p!\le |G|$?

Let $G$ be a finite group and $n_p:=|\text{Syl}_p(G)|$. Is it true that $n_p!\le |G|$ ? I've shown that it's true, but I'm not so sure, can you check my proof? Proof. Let $G$ act on ...
2
votes
0answers
41 views

Is there an infinite graph that corresponds to a group which has precisely all finite groups as subgroups?

This is a followup question to Pavel C's question here . It's fairly obvious from the axiom of choice that taking the direct sum of all finite groups produces the desired group. At the associated ...
2
votes
1answer
18 views

Prove that group $G$ is abelian when $K$ field has only 2 elements

Let $K$ be a field and $G$ is a group. $G=\{(g,a) : g\in K, a \in K^*\mid (g,a)(h,b)=(g+ah,ab)\}$ $K^*$ means $K$ without ${0}$. Proove that $G$ is Abelian $\Leftrightarrow$ $K$ has only 2 elements. ...
0
votes
1answer
27 views

Group Action and Orbits

I am looking at the following example which says find the orbit of $0$ under addition by $2$ and $3$ if $\mathbb{Z}_4$ acts on itself by addition. So to find the orbit of $0$ we are looking at the set ...
9
votes
0answers
109 views

Is there a group which has precisely all finite groups as subgroups?

I would like to ask the following question: Does there exist a group $G$ such that every finite group can be embedded in $G$, and every proper subgroup of $G$ is finite? The closest ...
2
votes
0answers
38 views

$G/H$ contains element of order $n$ but $G$ does not

I'm trying to come up with a group $G$ and normal subgroup $H$ of $G$ such that $G/H$ contains an element of order $n$ (for some integer $n$), but $G$ does not. Does $G = \mathbb{Z}$ and $H = ...
2
votes
2answers
30 views

Solution verification: $G$ and $G/H$ contain elements of same order

I just took my abstract algebra midterm, and was wondering if someone could confirm my solution to the following problem. Problem: Let $G$ be a finite group and let $H$ be a normal subgroup of ...
1
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1answer
18 views

How to prove 120 degree rotations of a hexagon form a subgroup

Let H={$\rho_{0}, \rho_{2}, \rho_{4}$}, a subgroup of D6, the group of symmetries. Where $\rho_{0}$=identity permutation, $\rho_{2}$=(1,3,5)(2,4,6) and $\rho_{4}$=(1,5,3)(2,6,4) Identity is easy to ...
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votes
1answer
33 views

Algebra: groups

Let m and n be two positive integers with gcd(m, n) = 1. Prove that Z/mZ × Z/nZ is congruent to Z/mnZ. [Consider φ : Z → Z/mZ × Z/nZ given by φ (a) = (a mod m, a mod n).] My Solution: Consider φ : Z ...
1
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1answer
22 views

Determine all the generators of $\mathbb{Z}_{25}^{\times}$

Determine all the generators of $\mathbb{Z}_{25}^{\times}$. Is there some way that I can use the fact that $\mathbb{Z}_{25}^{\times}$ is cyclic generated by $3$?
0
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1answer
31 views

Question on abstract algebra about Group?

I need an explanation, why $ (\mathbb{Z}_7,\oplus _6 )$ is not a Group? As I have discovered so far. The following conditions are satisfied I) Closed! II) Associative! III) ...
2
votes
1answer
25 views

Algebra (group theory)

Prove, without using Cauchy’s Theorem, that any finite group $G$ of even order contains an element of order two. [Hint: Let $S = \{\,g ∈ G : g \ne g^{−1}\,\}$. Show that $S$ has even number of ...
1
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1answer
29 views

Example of such groups [duplicate]

Does there exist $G$ such that for a subgroup $H$ of $G$ , $gHg^{-1}$ is proper in $H$ for some $g\in G$ ? It is clear that $H,G$ must be infinite. I look for examples in matrice groups and not ...
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votes
1answer
24 views

centralizer of a chief factor

Let $G$ be a finite solvable group and $p$ be a prime. Let $G^*$ be the smallest normal subgroup of $G$ for which the corresponding factor is abelian of exponent dividing $p-1$. Show that every chief ...
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votes
1answer
28 views

Abstract Algebra Groups of Order 2p

Groups of order 2p, where p is an odd prime. Suppose that G an element of order 2p. Prove that G isomorphic to Z2p. Hence G is cyclic. I can not use Sylow's theorem though since it has not yet been ...
2
votes
0answers
43 views

$2$-groups with odd permutations

If $P$ is the Sylow $2$-subgroup of a finite group $G$, $H <P$, and $x \in P$ so that no non-trivial element of $\langle x \rangle$ conjugates into $H$ (in $G$), and $|P|=|H||x|$, how can I show ...
2
votes
0answers
12 views

A relationship between hypercentral groups and abelian groups

Let $G$ be a hypercentral group. Suppose that $G$ is generated by a finite number of Prufer subgroups. Then $G$ is abelian? Remark: (1) $G$ is a hypercentral group if $G$ has a ascending central ...
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vote
4answers
54 views

If all Subgroups are Cyclic, is group Cylic?

I am having difficulty seeing if it is the case that if all subgroups of a group are cyclic, that the group itself is cyclic.
0
votes
0answers
13 views

A solid proof that Order of an r-cycle is r

This problem is so obvious and trivial for me that I don't know how to prove it. http://www.math.uiuc.edu/~weichsel/317sum01hwsols.pdf page 114,2.19 ...
1
vote
3answers
51 views

Does every group homomorphism from $(0,\infty),\times)$ to $(\mathbb{R},+)$ send $1$ to $1$?

I just have some true/false questions I am revising with and I'm not sure about this. Let $f:((0,\infty),\times)\to(\mathbb{R},+)$ be a group homomorphism, then $f(1)=1\tag{1}$ I know that a group ...
1
vote
0answers
44 views

Tannaka reconstruction: reference request

What is a classical and perhaps even original reference for the following result, often called Tannaka reconstruction? Let $G$ be a group and $R$ be a commutative ring in which $0,1$ are the only ...
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votes
0answers
89 views

Exercise on a quotient group over $\mathbb{Z}^2$

Let $G = \mathbb Z \times \mathbb Z = \mathbb Z ^2$ and let $H$ be the subgroup generated by $(1, 3)$ and $(2, 1)$, i.e.: $$H = \{m(1, 3) + n(2, 1) : m, n\in\mathbb Z\}.$$ This exercise will help ...