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.

learn more… | top users | synonyms (2)

2
votes
2answers
34 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 ...
1
vote
1answer
122 views

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

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 ...
2
votes
3answers
213 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
votes
1answer
76 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 ...
3
votes
1answer
61 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 ...
1
vote
1answer
21 views

Determining homomorphisms

I am asked to determine if the following mapping is a homomorphism: Define $\phi\colon D_4 \to \mathbb{Z}_4$ by $\phi(r^kf) = k$. I have deduced that $\phi(f) = 0$ as $\phi(f) = \phi(r^0f) = 0$. ...
0
votes
2answers
969 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?
1
vote
1answer
37 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. ...
4
votes
1answer
339 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
37 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
113 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
25 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
65 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 ...
15
votes
1answer
388 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 ...
3
votes
1answer
58 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
57 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 ...
2
votes
1answer
47 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 ...
1
vote
2answers
218 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
vote
1answer
60 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 $3$?
1
vote
1answer
43 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) ...
3
votes
1answer
37 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
vote
1answer
38 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 ...
-1
votes
1answer
49 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 ...
0
votes
1answer
65 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
51 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
29 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 ...
1
vote
4answers
317 views

If all Subgroups are Cyclic, is group Cylic? [duplicate]

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.
1
vote
3answers
69 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 ...
2
votes
1answer
106 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 ...
0
votes
1answer
130 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
45 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
99 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
67 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 ...
1
vote
1answer
556 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
0
votes
1answer
37 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 ...
3
votes
1answer
72 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
78 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
109 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 ...
2
votes
1answer
59 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.
0
votes
1answer
64 views

Isometry group of regular octagon, find commutating elements and 2 specific subgroups.

In isometry group of regular octagon $D_{16} = < g,s | \ g^8=1=s^2, sgs=g^{-1}>$ a) Describe a set of elements commutating with all elements of the group. b) Find 2 subgroups $A,B \subset ...
1
vote
0answers
88 views

A Mistake in GTM 247 (Braid Groups)?

I am reading Braid Groups (GTM 247) by Kassel Christian and Turaev Vladimir and am puzzled by a detail in the proof of a theorem: I do not quite see the reason of the inequality sign in the ...
1
vote
1answer
49 views

Finite-dimensional, irreducible Representations of the Diffeomorphism Group $Diff(R^4)$

Is there any possible way to study the finite-dimensional, irreducible representations of $Diff(R^4)$ systematically? My interests stems from the fact, that the symmetry group of general relativity is ...
1
vote
1answer
34 views

Why is this cyclic subgroup normal?

I'm looking at the following solution: I'm just wondering...how do we know that $\langle x \rangle$ is normal?
1
vote
2answers
49 views

Two questions regarding cyclic groups

We know that if $G$ is cyclic group of order $n$, then it is isomorphic to $Z_n$. I have two questions regarding this Does every finite abelian group have to be isomorphic to $Z_n$ for some $n$? If ...
3
votes
1answer
50 views

Simple or not simple Groups

True or False. Every subgroup of a simple group is itself simple. Our solution:The statement is false. As a counter example, let $G$ be equal to the group of even permutation of $10$ letters and $H$ ...
2
votes
1answer
77 views

Cyclic $G/Z(G)$ implies that $G$ is abelian [duplicate]

Suppose that $G$ is a group where $|G| = pq$ for prime $p$ and $q$. Suppose further that $|Z(G)| = q$. Thus, $Z(G) \cong Z_q$ (it is a cyclic group). Moreover, we see that $$ |G/Z(G)| = ...
2
votes
2answers
89 views

Prove that $gNg^{-1} \subseteq N$ iff $gNg^{-1} = N$

I'm reading the solution to one of my homework problems and am stuck on something. Here is the problem: Let $N \leq G$ be a finite subgroup. Show that $gNg^{-1} \subseteq N$ if and only if ...
1
vote
1answer
37 views

Isometry group question

The isometry group of $\mathbb{R}^n$, $\def\Isom{\operatorname{Isom}}$ \begin{eqnarray*} \Isom_n(\mathbb{R})=\{f\colon \mathbb{R}^n \to \mathbb{R}^n: f ~ \text{is an isometry}\}. \end{eqnarray*} How ...
2
votes
1answer
61 views

In the definition of a quotient group, does the subgroup have to be normal?

Let $G$ be a group and let $H \leq G$. Does $H$ need to be a normal subgroup to have the quotient group $G/H$? Progress I think yes. By definition, a subgroup $H$ is normal if and only if it is ...
1
vote
1answer
23 views

Constructing a homomorphism from $G$ to $S_t$ and finding it's order

Let $H$ be a subgroup of $G$ with index $t$. Let $X$ be the set of left cosets of $H$ in $G$, so that $|X| = t$. Then $G$ acts on $X$ by left translation: $g · (xH) := (gx)H$. (a) Use this to ...