Tagged Questions

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

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0answers
14 views

Access for GAP (Maths)

Since 1992, sets of user contributed programs, called packages, have been distributed with GAP. For convenience of the GAP users, the GAP Group redistributes packages, but the package authors remain ...
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
29 views

Order of a Group with certain elements of composite order

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
34 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
26 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
18 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?
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3answers
44 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
26 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
39 views

Simple group with Klein four

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
53 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
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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
88 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
53 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
19 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
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1answer
38 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 ...
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2answers
28 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?
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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
14 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
25 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
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0answers
34 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
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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. ...
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0answers
23 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 ...
8
votes
0answers
92 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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1answer
32 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
votes
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
24 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
28 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
27 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
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0answers
42 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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4answers
53 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
50 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
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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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1answer
88 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 ...
1
vote
1answer
24 views

Prove that $C_H(K) = N_H(K)$ for $G=H \rtimes_{\phi} K$

Let $H,K$ be group where $\phi: K \rightarrow \operatorname{Aut}(H)$ is a homomorphism. Also, let $G=H \rtimes_{\phi} K$. Show that $C_H(K) = N_H(K)$ Proof: Let $h\in N_H(K)=\{h\in H: ...
1
vote
1answer
50 views

Find subgroups $H$ and $K$ of $G$ such that $H$ is isomorphic to $G$ but $G/H$ is not isomorphic to $G/K$

Let $G=\mathbb{Z}_2 \times \mathbb{Z}_4$, 1). Find subgroups $H$ and $K$ of $G$ such that $H$ is isomorphic to $K$ but $G/H$ is not isomorphic to $G/K$. 2). Find subgroups $A$ and $B$ of $G$ such ...
0
votes
1answer
28 views

Find a permutation with the given square or cube

Problem: find a permutation such that $x^2 = (1\;3\;4\;5\;7)$, $x\in S_7$ $x^3 = (1\;3\;4\;5\;7)$, $x\in S_7$ Must find all possible solutions for $x$. Progress I have solved for the first ...
0
votes
1answer
38 views

list the distinct principal ideals in $\mathbb{ℤ}_2 \times \mathbb{ℤ}_3$

How do I find and list the distinct principal ideals in ℤ2xℤ3? I know that Z2 has 0,1 and that Z3 has 0,1,2, but I'm not sure how to list them and how to find ideals in Z2xZ3
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1answer
27 views

Orders of Cosets

Let H be a normal subgroup of G and for any $g \in G$ that $|g|=n$ and $|G/H|=m$. Suppose the $gcd(n,m)=p$ where $p$ is prime. Show that for any $ a\in gH$, then $a^p\in H$. Could I get some help on ...
2
votes
1answer
32 views

A unique operation on a set that makes it a group

From Rotman's "Introduction to Group Theory": Let $G$ be a group and let $X$ be a set having the same number of elements as $G$. If $f:G\rightarrow X$ is a bijection, there is a unique binary ...
3
votes
1answer
27 views

Number of $k$-cycles in $S_n$

I've computed that the number of $k$-cycles in $S_n$ is $\frac{n!}{(n-k)!k}$ and wiki seems to agree with me. Now, we know that in $S_n$ the number of $k$-cycles is also equal to the cardinality of ...
0
votes
1answer
34 views

Generator of $Z_p^*$ with large p

I have to find a generator for $Z_{p}^*$. The prime number p is $2425967623052370772757633156976982469681$. My prime factors for (p-1) is according to 1 ...
1
vote
1answer
35 views

The subgroups of $D_6$ of order $2$

Let $D_6=\{e,r,r^2,r^3,r^4,r^5,a,ar,ar^2,ar^3,ar^4,ar^5\}$ where $r^6 = 1$ and $a^2=1$. I am confused as to how we find the subgroups of order $2$ other than the center subgroup.