Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [group-theory]

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 is the study of groups.

0
votes
0answers
11 views

Normal group of abelian and centerless groups product

I am trying to understand the structure of normal groups of $G=A\times H$ where $A$ is an abelian group and $H$ is a centerless group (both are finite). I want to show that if $N \vartriangleleft G$ ...
1
vote
1answer
24 views

Proof of coset/subgroup property

I am given a proof of the following property: The only left coset of $H$ which is a subgroup of $G$ is $H$ itself. The brief proof is based on the following properties: 1)  $xH = H \iff x\...
0
votes
1answer
28 views

How can I prove that $H_1/N_1 \ncong H_2/N_2$?

So I have a non-surjective homomorphism $\phi: H_1 \to H_2$, and $N_1 \unlhd H_1$ and $\phi(N_1) = N_2$. How do I prove that $H_1/N_1 \ncong H_2/N_2$? All I have right now is an example that works, ...
0
votes
0answers
35 views

Is $H/N$ a quotient group even if $N$ is just a subgroup of $H$ and not necessarily normal?

I am very confused about the notation of $H/N$ and whether it always implies a quotient group. The confusion stems from the following statement of the fourth isomorphism theorem: Let $G$ be a group ...
3
votes
0answers
31 views

Automorphisms “killing” and group

I ran into the following concept in passing here. Let $G$ be a group and let $\phi$ be an automorphism of $G$. Let $P$ be a presentation for $G$ with $X$ the set of generators in $P$. Form a new ...
4
votes
2answers
106 views

Does every subgroup of an abelian group have to be abelian?

My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such ...
0
votes
0answers
26 views

Detect if an element is in an orbit.

Suppose $n = pq$ is a product of two distinct primes. We also know that integer $r$ divides $\varphi(n) = (p-1)(q-1)$ and $r^2$ doesn't divide it. Fix element $x$ of $\mathbb{Z}_{n}$. Can we tell ...
1
vote
1answer
21 views

Determine whether the statement is true: “If $g\in G$ has finite order $|g|=m$ then $f(g)$ has order $m$ in $H$.”

Determine whether the following statement is true or false: "Suppose $f : \{G, *\} \mapsto \{H, \circ \}$ is a homomorphism of groups, and let $f(G) = \{f(g)~|~g\in G\}$. If $g\in G$ has finite order $...
1
vote
2answers
32 views

Find the smallest $n \in \mathbb{N}$ such that the group is isomorphic to the direct product of $n$ cyclic groups

Find the smallest $n \in \mathbb{N}$ such that the group $\mathbb{Z}_{6} \times \mathbb{Z}_{20} \times \mathbb{Z}_{45}$ is isomorphic to the direct product of $n$ cyclic groups. I'm not sure but if I ...
2
votes
0answers
25 views

What does Sylow theory have to say about group presentations?

What does Sylow theory have to say about group presentations? Of the books on combinatorial-group-theory I have looked in so far, the following do not contain any reference to Sylow's Theorems: ...
9
votes
0answers
77 views

If $H_1 \subset H_2 \subset G$ and $G/H_2,\ H_2/H_1$ are compact then $G/H_1$ is compact.

I'm trying to solve the following exercise of the book "Grupos de Lie - Luiz A. B. San Martin (exercise 18, page 55)": Exercise: Let $G$ be a topological group and $H_1 \subset H_2\subset G$ ...
1
vote
0answers
16 views

The multiplicative group of units in $\mathbb{Z}_{p}$ is isomorphic to $\mathbb{Z}_{p} \times C_{n}$ with $n=\max\{p-1,2\}$.

Problem. (a) Write down explicitly the rules for addition and multiplication in $\mathbb{Z}_{p}$. (b) Show that $(\mathbb{Z}_{p^{i}},\varphi_{ij})$ where $\varphi:\mathbb{Z}_{p^{j}} \to \mathbb{...
1
vote
1answer
36 views

Groups containing finite index subgroups isomorphic to $\mathbb{Z}$

There are two propositions I'm trying to prove that seem fairly intuitive to me, but I haven't been able to prove: Let $G = A \ast B$ with $A,B$ infinite. No subgroup of $G$ with finite index is ...
0
votes
1answer
21 views

Surjectivity of a bilinear map between groups

Let $G$ a generic group and $\gamma_i(G)$ its lower central series defined as $$ \begin{cases} \gamma_1(G)=G\\ \gamma_{i+1}(G)=\left[\gamma_i(G); G\right] \end{cases} $$ for every $i\in \mathbb N$. ...
1
vote
2answers
35 views

If $g^5 = h^7$ in a free group, then $g$ and $h$ are in a cyclic subgroup

Let $F$ be a free group, $g,h \in F$ with $g^5 = h^7$. Then I want to show these are in a cyclic subgroup. The strategy I'm trying and failing with is to show that $g$ and $h$ commute, then they are ...
0
votes
1answer
33 views

Showing a subset is a subgroup

Let $A$ be a subset of a finite group $G$. For any $g\in G$ we denote by $gA$ the set $\{ga : a ∈ A\}$. Assume that $e \in A$, where $e$ is the identity element of $G$, and that for any $g_1, g_2 \in ...
-1
votes
1answer
24 views

Show that if N, K are normal subgroups of a group G, and N contains K then we have: $ G / N \cong (G/K) / (N /K) $ [duplicate]

Show that if $N, K$ are normal subgroups of a group $G$, and $N$ contains $K$ then we have: $$ G / N \cong (G/K) / (N /K) $$ Intuitively it looks correct, would like to know how I can approach this.
2
votes
2answers
57 views

The number of groups $G$ such that $G/\mathbb{Z}_3\cong D_{2n}$

I am trying to find the number of groups $G$ such that $G/\mathbb{Z}_3\cong D_{2n}$, where $\mathbb{Z}_3$ denotes the cyclic group of order $3$ and $D_{2n}$ denotes the dihedral group of order $2n$. ...
6
votes
1answer
83 views

Number of subgroups of an abelian p-group

Let $p$ be a prime number and let $n\in \mathbb{N}$. I know that every abelian group of order $p^n$ is uniquely a direct sum of cyclic groups of order $p^{\alpha_i}$ where $\sum \alpha_i = n$. Now the ...
2
votes
1answer
60 views

Let $S=\{x~|~x\in G$ and $x^2 \in H\}$. Show that $S$ is a subgroup of $G$ for $H<G$, $G$ abelian.

The question states: Let $G$ be an Abelian group with subgroup $H < G$. Let $S=\{x~|~x\in G$ and $x^2 \in H\}$. Show that $S$ is a subgroup of $G$. My proof is different than what is in the ...
2
votes
0answers
32 views

Size of conjugacy class of a quotient group

I am trying to recall what happened to the conjugacy class and centraliser when we quotient out by a normal subgroup. In particular, we know from Orbit-Stabiliser that $|G|=|ccl_G(g)||C_G(g)|$ for ...
7
votes
2answers
114 views

Construct a nonabelian group of order 44

Let $G$ be a group s.t. $|G|=44=2^211$. Using Sylow's Theorems, I have deduced that there is a unique Sylow $11$-subgroup of $G$; we shall call it $R$. Let $P$ be a Sylow $2$-subgroup of $G$. Then we ...
0
votes
1answer
58 views

Working on Galois Theory. Splitting field of f(x) = $x^2 + 11 \in Q[x]$

Working on Galois Theory. I have a polynomial of f(x) = $x^2 $+11$\in$ Q[x] and I am asked to find the splitting field. I know that solutions to f(x) = 0 are $i\sqrt 11$ and $-i\sqrt 11$. I also ...
-3
votes
0answers
33 views

Stabilizer, Orbit, Coset, Normalizer [on hold]

The group $G = D_4$ is the group of symmetries of the square $X$ with vertices $A, B, C, D$ (in cyclic order). Let $D_4$ act on the set S = { $A$, $B$, $C$, $D$, $\overline{AC}$, $\overline{BD}$}. ...
1
vote
1answer
32 views

Show that an infinite abelian group all of whose proper quotients are finite $\cong \mathbb{Z}$

Show that an Infinite abelian group all of whose proper quoteints are finite $\cong \mathbb{Z}$ So I'm a little confused on how to get started here. Perhaps something like: Let $x \in G$, then $|G/&...
1
vote
0answers
29 views

Let $G$ be a group s.t. $|G|=pqr$ where $p,q,r$ are primes that need not be distinct. Prove that $G$ is soluble.

Let $G$ be a group s.t. $|G|=pqr$ where $p,q,r$ are primes that need not be distinct. Prove that $G$ is soluble. So, I don't know whether I should handle this case by case and try to get the Sylow ...
2
votes
2answers
31 views

Is it necessary that $f: G_1 \to G_2$ is an isomorphism, for $f:H\to H$ an automorphism with $G_1, G_2\le H$ of the same cardinality?

This might be a very trivial question, and I have explained what I think about it below. Let's say I have an automorphism $f : H \to H$. Now, let's say I take two subgroups $G_1$ and $G_2$ of $H$. Is ...
1
vote
1answer
37 views

RSA - statements re $\phi(n)$, $\lambda(n)$ and $n$, given $n$ and $e=65537$

Assume $n=pq$, with $p,q$ primes, $e=65537$, and length of $n$, $|n|=N=1024$ bits = 309 decimal digits. $p,q$ are unknown. I am trying to understand the information sourced from Wikipedia page on RSA ...
0
votes
0answers
15 views

Prove: if group $G$ has only one element $a$ of order $n \neq 1, n \in \mathbb{N} $ , then $ a \in C(G)$ and $n = 2$ [duplicate]

Prove: if group $G$ has only one element $a$ of order $n \neq 1, n \in \mathbb{N} $ , then $ a \in C(G)$ and $n = 2$ , where $C(G)$ is the center of group G. I think I proved $ a \in C(G)$ for an ...
0
votes
1answer
18 views

number of permutation of S4 as product of two disjoint cycles each of length 2

There was a problem of finding out the number of permutations of order 2 in S4. There are two cases. case-1 permutation of single cycle of length 2. case-2 permutations of two disjoint cycles ...
1
vote
2answers
41 views

Quotient group of dihedral group

Let $G=\{e,r^{2},...,r^{8},s,sr,...,sr^{8}\}$ and let $N=\langle r^{3} \rangle.$ Now let $\pi(g)=\bar{g}=gN$ be surjective with kernel $N$. I have to show that $G/N=\{\bar{e},\bar{r},\bar{r^{2}},\...
2
votes
0answers
52 views

Dynamical systems from an algebraic perspective

I am interested in learning more about dynamical systems. Most of my background is algebraic, specifically in group theory/geometric group theory. I was wondering if anyone knew of a reference that ...
2
votes
1answer
22 views

Intersection of connected components with discrete subgroup

I am currently studying Harmonic Analysis and didn't quite understand part of the proofs for the structure theorems of locally-compact abelia groups (LCA). Let $A$ be an LCA group such that $\hat{A}/\...
1
vote
2answers
55 views

Does $\langle ( 1 ,3), (1, 2 … ,10)\rangle$ generate the group $S_{10}?$ [duplicate]

Does $\langle ( 1, 3), (1 ,2 ..., 10)\rangle $ generate the group $S_{10}$ ? I think that's it doesn't because every use of $(1, 3)$ makes a "jump" between at least two numbers. So we can get for ...
1
vote
1answer
37 views

Is there an easy expression for multiplicative inverses in $\mathbb Z_p$?

I know that in arbitrary division rings, one can go about finding inverses Euclidean division. But take $\mathbb Z_{11}$ as a simple example. Is there a "nice" expression which yields the inverses in ...
0
votes
1answer
20 views

A finite non-abelian group of order $n$ that for every divisor of $n$ has a subgroup is not simple

Let $G$ be non-abelian group of order $n$. Also, for every $k$ which is a divisor of $n$ , there is a subgroup of $G$ of order $k$. I want to prove that $G$ is not simple. Well, from what is given, I ...
0
votes
2answers
46 views

A proof about Automorphism in congruence class

Suppose $gcd(m,n)=1$, and let $F :Z_n→Z_n$ be defined by $F([a])=m[a]$. Prove that $F$ is an automorphism of the additive group $Z_n$. I find it is diffcult to prove $F$ is injective and surjective. ...
3
votes
1answer
186 views

Is there a canonical “inverse” of Abelianization?

We know that the abelianization of a free product is the direct sum, for example $Ab(\mathbb{Z}*\mathbb{Z})=\mathbb{Z}\oplus\mathbb{Z}$. Is there a “canonical” (or even non-canonical) operator that ...
1
vote
0answers
21 views

Using Zassenhaus to obtain an isomorphism of bigraded groups

Let $G$ be a group. Suppose that $F^{\bullet}G$ is a filtration on $G$. If $q: G \to K$ is a quotient map, then we get an induced filtration of $K$ given by $F^aK:= q(F^aG) = (F^aG)/(F^aG \cap \ker q)$...
1
vote
3answers
55 views

Suppose that $H_1/H_2$ is Abelian. Show that $H_1 N / H_2 N$ is Abelian.

Suppose $G$ is a group and $H_1$, $H_2$, $N$ are subgroups of $G$. $N$ is a normal subgroup of $G$ and $H_2$ is a normal subgroup of $H_1$. Suppose that $H_1/H_2$ is Abelian. Show that $H_1N/H_2N$ is ...
1
vote
3answers
36 views

If $N$ is a subgroup of $G$ and $M$ is a normal subgroup of $G$, show that $NM$ is a subgroup of $G$.

I am just learning to work with normal subgroups, and am trying to do the following proof. However, I have had some problems. Let $N$ and $M$ be subgroups of a group $G$, where $M$ is a normal ...
0
votes
1answer
31 views

Suppose G is a group, a, b ∈ G such that |b| = 2 and bab = a^4 . [on hold]

I know that answer is (2) |a| divides 15 but I'm not sure how they got the answer, any hints will be helpful thank you.
1
vote
0answers
36 views

$H \cong \frac{Gl_2(\mathbb{R})}{SO(2)}$ [duplicate]

I am trying to prove that the upper-half complex plane H, is isomorphic to General Linear group of order 2 (over the reals), quotationed out by the special orthogonal group. I believe I can prove ...
2
votes
2answers
49 views

Computing the order of $[9]_{31}$ in $(\mathbb{Z}/31\mathbb{Z})^*$

A part of Aluffi's "Algebra: Chapter 0" exercise II.4.12 suggests computing the order of $[9]_{31}$ in $(\mathbb{Z}/31\mathbb{Z})^*$. Sure, I could just multiply $9$ a few times until I get $1$ as a ...
1
vote
1answer
32 views

Elements conjugate in profinite completion

Problem. Let $G$ be a residually finite group, and identify $G$ with its image under the canonical map to its profinite completion $\hat{G}$. Let $x,y \in G$. Prove that the following conditions are ...
1
vote
1answer
41 views

Number of generators of subgroup

I am trying to prove the following. let $G$ be a finitely generated abelian group, and $H<G$ a subgroup such that there exists a subgroup $K<G$ and we can write $G=H \oplus K$. Is it true that ...
1
vote
0answers
29 views

The coset corresponding to permutation $(123)$ in $\Bbb Z /3\Bbb Z$.

We know that $V_4 \triangleleft A_4$ so $ A_4/V_4 \cong \Bbb Z /3\Bbb Z$. The coset corresponding to permutation $(123)$ is $(123)V_4$. Is it corresponding to $\overline{1}$ or $\overline{2}$ in $\...
0
votes
0answers
21 views

Question about primitive roots and multiplicative groups

Let $p$ be an odd prime and let $k \in \mathbb{N}$. We know that $U(\mathbb{Z}_p) = \{\overline{1},\overline{2},\dots,\overline{p-1}\}$. Can it happen and when that $U(\mathbb{Z}_p) = \{\overline{1}^k,...
2
votes
1answer
26 views

Use the second isomorphism theorem to show that (𝑁∩𝐻)/(𝑀∩𝐻) is Abelian.

Suppose that 𝐻,𝑁,𝑀 are subgroups of 𝐺, 𝑀 is a normal subgroup of 𝑁. Assume 𝑁/𝑀 is Abelian. Use the second isomorphism theorem to show that (𝑁∩𝐻)/(𝑀∩𝐻) is Abelian. I can show (𝑁∩𝐻)M is ...
3
votes
0answers
38 views

Minimal order of a counterexample to Wall’s conjecture

There used to be once a rather well known and interesting conjecture, that was formulated by Gordon E. Wall: The number of maximal subgroups of a finite group $G$ does not exceed $|G|$ That ...