Tagged Questions

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

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2
votes
2answers
31 views

Free Group generated by S is actually generated by S.

Consider the definition of free groups via the universal property: Definition. We say that the group $F$ is the free group generated by the set $S$ if there's a map $f:S\to F$ such that whenever ...
2
votes
1answer
22 views

The Fricke involution on arbitrary subgroups of $\operatorname{SL}_2(\textbf{Z})$

Let $\Gamma$ be a subgroup of $\operatorname{SL}_2(\textbf{Z})$. I want to understand how the Fricke involution $w_N : f \longrightarrow (\tau \longrightarrow N^{-k/2}\tau^{-k}f(-1/N\tau))$ acts on ...
-2
votes
0answers
30 views

Group with some property

What properties next group has? $G= \left\langle\begin{pmatrix}1+pa &0\\pb& 1+pc\end{pmatrix}\right\rangle\left\langle\begin{pmatrix}1+pe &pr\\0& ...
3
votes
1answer
35 views

Endowing an abelian group with a metric.

I solved the following exercise, which is not hard: Let $G$ be an additive abelian group, such that exists $f: G \to \mathbb{R}$ satisfying: $f(0) = 0$ and $f(x) > 0$ for all $x \neq ...
0
votes
0answers
28 views

Orbit Stabilizer Theorem? Question on HW? [duplicate]

I have 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 $A_1$ ( ...
1
vote
0answers
9 views

Calculating dihedral 6 factor group

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). I am trying to ...
3
votes
1answer
14 views

Number of preimages of a group element under a homomorphism.

I was reading the first chapter of Robert Gilmore's "Lie groups, physics and geometry" and I came across a brief statement regarding the number of preimages of an element under an homomorphism which I ...
0
votes
0answers
15 views

Prove that a finite group is polycyclic if and only if solvable [on hold]

I'm learning group theory but I'm not sure how to solve this problem.
1
vote
3answers
22 views

Find all the cosets of Dihderal group 6 with subgroup H

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). How would you ...
0
votes
0answers
28 views

Algebra: groups and closets

We know P = {λ (1, 3) + μ(2, 1) : 0 ≤ λ , μ < 1}. Now let $(x, y) ∈ Z ^2$. Because (1, 3) and (2, 1) span $R^2$ there are real numbers s and t such that (x, y) = s(1, 3) + t(2, 1). Let (a, b) = (s ...
3
votes
0answers
37 views

Group Theory Research Topics [advice]

I'm currently a Senior Mathematics student in the US. I'm interested in Abstract Algebra, specifically Group Theory (I've taken a course that dealt entirely with Fields, but I enjoy Groups a bit more) ...
0
votes
1answer
24 views

Can someone describe SO(n)/SO(n-1) for me?

I don't have a ton of experience with group theory, but this came up in a topology class and I was just wondering if someone could give me an intuition for this quotient space, and describe what group ...
0
votes
1answer
21 views

Verification of proof that NM is a normal subgroup of G if M and N are both normal subgroups of G

My proof is as follows: M is a subgroup of G means $g_1mg_1^{-1}$ is part of M and likewise N is a subgroup of G means $g_2mg_2^{-1}$. To prove our claim do the following: $$g_1mg_1^{-1} ...
1
vote
0answers
14 views

Verification of Proof that if N is a normal subgroup and H is any subgroup HN={hn| h in H and n in N} is a subgroup

My proof is as follows: I only have question for the closure portion. Closure: Let h1, h2 be in H and n1, n2 be in N. Since N is normal we can say $Nh_2=h_2N$. This also means $n_1h_2=h_2n_3$ So ...
-1
votes
0answers
28 views

Question about the Orbit Stabilizer Theorem? [duplicate]

I've proved that $H = \{f_m,0 \mid m \in \mathbb{R}^*\}$ is a non-normal subgroup of $A1$ ( the group of affine functions, $f=mx+b$). I've also proved that the right cosets of $H$ are all in the ...
3
votes
2answers
30 views

Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center

A nonabelian group $G$ of order $pq$ where $p$ and $q$ are primes has a trivial center My Proof is as follows: Assume we have nonabelian group $G$ of order $pq$ where both $p$ and $q$ are ...
1
vote
2answers
42 views

If $G/Z(G)$ is cyclic then $G$ is abelian – what is the point?

The theorem "if $G/Z(G)$ is cyclic then $G$ is abelian" is a popular exercise. But what is the point of this theorem if $G/Z(G)$ can only be cyclic if it is trivial? Does "$G/Z(G)$ is cyclic" ...
1
vote
1answer
26 views

Verification of Proof that if G is not abelian G/Z(G) is not cyclic

I will prove this by the contrapositive: "If G/Z(G) is cyclic then G is abelian" Proof: We assume that G/Z(G) is cyclic. This means it is generated by a left coset $(aZ(G))^n$=e for some integer n. ...
-4
votes
0answers
41 views

Group of Homomorphism [on hold]

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
vote
1answer
49 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 ...
2
votes
1answer
31 views

Fast way of finding a homomorphism

Show that $D_{12}$, the dihedral group of order $12$, 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 ...
0
votes
0answers
64 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
votes
0answers
20 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 $G$ ...
-2
votes
2answers
19 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 ...
1
vote
0answers
31 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
votes
1answer
32 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
votes
1answer
33 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
votes
2answers
25 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
votes
1answer
19 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
vote
0answers
40 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
vote
1answer
38 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
votes
0answers
31 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 ...
0
votes
0answers
21 views

What is the structure of the group with at most five elements of composite order? [on hold]

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
vote
3answers
50 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
votes
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
votes
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
51 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
65 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
24 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 ...
2
votes
2answers
117 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
22 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
43 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
30 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
16 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
27 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
45 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
28 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 ...