Tagged Questions

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

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4
votes
4answers
351 views

On order of elements of a infinite group

Let $a$ and $b$ be elements of finite order of an infinite group $G$. Then do we can say that order $ab$ is finite? I think this is true for Nilpotent Groups.
1
vote
2answers
206 views

Class equation, center and commutator subgroup

I'm trying to determine the class equation, center and commutator subgroup of the group $G=\langle x, y\mid x^4,y^4,yxy^{-1}x\rangle$ First of all, I tried to determine $|G|$ directly. I could ...
6
votes
2answers
887 views

Classify groups of order 27

Let $|G|=27$. Prove that all subgroups of index 3 are normal. Classify all groups of order 27. I can do the first one, but the classification is overwhelming. I don't even know where to start. ...
5
votes
3answers
363 views

Is $(\mathbb{Q},+)$ the direct product of two non-trivial subgroups?

Is this statement true or false? I am really not having any idea how to prove or a counterexample, please help. Is $(\mathbb{Q},+)$ a direct product of two non-trivial subgroups?
2
votes
1answer
183 views

Mapping between subgroups by an isomorphism.

The original question that I wrote was: A subgroup characteristic of the whole group I was wishing that there is an argument that is simple enough to see that "clearness" since the book that I am ...
4
votes
1answer
41 views

Does this theorem hold for the infinite case as well?

Consider the theorem: Orbits of a normal subgroup are equal in size when the full group acts transitively This is stated and proved here. I wonder: Does this theorem also hold infinite sets on ...
8
votes
1answer
494 views

Automorphism group of the elliptic curve $y^2 + y = x^3$

Consider the elliptic curve $E : y^2+y = x^3$ over $\overline{\mathbb{F}_2}$. It has the biggest automorphism group $G$ among all elliptic curves, namely with order $24$. What is the structure of $G$? ...
3
votes
0answers
85 views

On groups of order $6n$.

Let $G$ be a finite group such that $|\operatorname{Inn}(G)|=6$ and $Z(G)$ is an elementary abelian $2$-group of order $2^n$. Then prove or disprove that $G$ is isomorphic to ...
1
vote
0answers
47 views

Duality between $[G,G]$ and $Z(G)$? [duplicate]

Possible Duplicate: Center-commutator duality Let $G$ be a group. It seems that there is a certain duality between two of its normal subgroups, the commutator \begin{equation} ...
2
votes
1answer
98 views

A subgroup characteristic of the whole group

We say a subgroup $H \leqslant G$ is characteristic in $G$ if for every $\varphi \in \text{Aut}(G)$, we have $\varphi(H) = H$. Now, suppose that we have a unique subgroup $S \leqslant G$ that has a ...
21
votes
2answers
492 views

Lower bounds on the number of elements in Sylow subgroups

Let $p$ be a prime and $n \geq 1$ some integer. Furthermore, let $G$ be a finite group where $p$-Sylow subgroups have order $p^n$. Denote by $n_p(G)$ the number of Sylow $p$-subgroups of $G$. Denote ...
0
votes
1answer
61 views

Elements of order $9$ in $C_{135} \times C_6 \times C_9$

Ok, so I first split everything up into powers of primes and get $$C_{27} \times C_5 \times C_2 \times C_3 \times C_9$$ Now, I want to find elements of order 9 in each of these cyclic groups. This ...
3
votes
2answers
273 views

On the minimal number of generators of a finite group

Let $G$ be finite group and d(G) be the minimal number of generators of $G$ and $H$ be subgroup of $G$. Then prove or disprove $$d(H)\leq d(G)?$$ What about abelian groups? For example is true for ...
2
votes
2answers
265 views

$p$-group and normalizer

Here is the question: a) Show that if $p$ is a prime number and $P$ is a $p$-subgroup of a finite group $G$, then $[G:P]=[N_G(P):P]$(mod p), where $N_G(P)$ denotes the normalizer of $P$ in $G$. ...
1
vote
2answers
172 views

Question about group presentation and free group

Is it true that the group presented by $\langle a, b \mid ab = 1\rangle$ is isomorphic to the free group generated by $a$?
1
vote
2answers
101 views

Number of elements of order $45$ in $C_{45} \times C_{15} \times C_{10}$

If I had to do this question, can I say: $$C_{45} \times C_{15} \times C_{10} \cong C_{9} \times C_5 \times C_3 \times C_5 \times C_2 \times C_5$$ Now, as I want elements of order $45$, lets first ...
4
votes
2answers
418 views

A normal subgroup $H$ with $[G:H]$ coprime to $p$ contains every Sylow $p$-subgroup of $G$.

I was working on the following question: Let $p$ be a prime factor of the order of a finite group $G$. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that H must ...
1
vote
2answers
72 views

Show by example that this need not be true if we do not assume that the groups are finitely generated

Let $G, H,$ and $K$ be finitely generated abelian groups. If $G \times K \cong H \times K$, show that $G \cong H$. Show by example that this need not be true if we do not assume that the groups ...
9
votes
1answer
363 views

Involutions and Abelian Groups

Suppose that $ G $ is a finite group where at least three-fourths of the elements are involutions, i.e., $$ |I(G)| \geq \frac{3}{4} |G|. $$ (Here, $ I(G) $ denotes the set of all involutions of $ G $, ...
2
votes
1answer
82 views

Finding the order of $N_G(H)$

I have this question: For a group $G=A_5$ and $H=$ $\langle(12)(34),(13)(24)\rangle$ , prove that $(123)\in N_G(H)$ and hence deduce the order of $N_G(H)$. $A_5$ is defined to be the alternating ...
0
votes
1answer
31 views

Slight mistake in working out my semi direct product

Construct explcitly a non-commutative semi direct product $H \rtimes Q$ with $$H = C_{79} \hspace{2cm} Q = C_{13}.$$ You may assume that the least positive integer $k \geq 1$ such that ...
7
votes
3answers
341 views

Free group n contains subgroup of index 2

My problem is to show that any free group $F_{n}$ has a normal subgroup of index 2. I know that any subgroup of index 2 is normal. But how do I find a subgroup of index 2? The subgroup needs to have ...
0
votes
3answers
105 views

$M_1$ and $M_2$ are subgroups and $M_1/N=M_2/N$. Is $M_1\cong M_2$?

Let $G$ is a group and $N$ is normal in $G$ and $M_1$ and $M_2$ are subgroups which contain $N$ such that $M_1/N= M_2/N$. Can we deduce that $M_1$ and $M_2$ are isomorphic? Thank you for hints.
1
vote
2answers
755 views

If every cyclic subgroup of $G$ is normal so is every subgroup?

I searched the Web to find an conterexample for this statement: If every cyclic subgroup of group $G$ is normal in $G$ then every subgroup of $G$ is normal in $G$. But couln't find any. It seems it ...
3
votes
0answers
56 views

reference request: “p-adic” presentation of surfaces

On several occasions I heart about the following result: For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...
3
votes
1answer
207 views

On inner automorphisms group of semidirect product of two groups

Let $G$ and $H$ be two groups and $\phi: H \rightarrow Aut(G)$ be a group homomorphism . Then we know $$Inn(G\times H)=Inn(G)\times Inn(H).$$ What about $Inn(G \rtimes H)$? I think (not sure) we have ...
1
vote
3answers
631 views

Prove that that $U(n)$ is an abelian group.

Prove that that $U(n)$, which is the set of all numbers relatively prime to $n$ that are greater than or equal to one or less than or equal to $n-1$ is an Abelian group. My thought process: for $a, b ...
5
votes
4answers
2k views

What is the difference between a Subgroup and a subset?

What is the difference between a Subgroup and a subset? I know hardly any Abstract algebra, just some things from youtube and wikipedia, but the notion of a subgroup being part of a larger group and a ...
11
votes
1answer
290 views

Topological Meaning of semi-direct product

I know that the amalgamated free product of two groups $G\star_K H$ has a certain topological meaning. What about a semi-direct product $H \rtimes G$ ?
2
votes
1answer
138 views

Construct semi direct product $H \rtimes Q$

EDIT: I'm only allowed to ask 6 questions and as my next question is similar to this, I thought I'd post it in this thread. The first question is the new one. The one under the underlined bit is the ...
3
votes
1answer
81 views

List of finite groups of Lie type and their BN-pairs

as the title states I am looking for a list of classical groups (or perhaps finite groups of Lie type) and their respective BN-pairs (or isomorphism type of the respective Weyl group). A quick Google ...
6
votes
2answers
458 views

A field that is an ordered field in two distinct ways

Question: Explain the construction below (taken directly from Counter Examples in Analysis): An ordered field is a field $F$ that contains a subset $P$ such that $P$ is closed with respect ...
0
votes
1answer
90 views

Is my reasoning behind my construction of my semi direct product correct?

Classify up to isomorphism the groups of order $203$. Assume the face that the least $ k \geq 1$ such that $$2^k \equiv 1 \mod{29}$$ is $k = 28$. [HINT: Look at the Sylow subgroups and ...
1
vote
1answer
57 views

Multiplication law for my semidirect the same?

I constructed my semidirect product and got the multiplication law to be $$ab = ab^{64}$$ where as in the answers it said it was $$ba = ab^8.$$ I want to know if my answer was also correct by ...
0
votes
0answers
84 views

Construct explicitly a non-commutative semi direct product

If I am asked to "Construct explicitly a non-commutative semi direct product..", does that mean that because they have semd "non commutative", I don't include the trivial SDP, i.e the direct product? ...
0
votes
0answers
100 views

Construct a homomorphism

I try to construct a homomorphism $$\mathbb{Z}[x] \rightarrow \mathbb{Z}[D_p]$$ which is surjective and where $D_p$ is the dihedral group, $\mathbb{Z}[D_p]$ a monoidring . My problem is that $D_p$ ...
1
vote
2answers
116 views

What is $\mathbb{Z}/m\mathbb{Z}/n (\mathbb{Z}/m\mathbb{Z})$?

Is $\mathbb{Z}/m\mathbb{Z}/n (\mathbb{Z}/m\mathbb{Z})=\mathbb{Z}/(m\mathbb{Z} +n \mathbb{Z})$? Thanks.
0
votes
1answer
197 views

Problem in modular arithmetic using group theory

This problem is from Herstein's Topics in Algebra. I have to use the property that if a finite set is closed under an associative product and that both cancellation laws hold in $G$, then $G$ is a ...
4
votes
2answers
103 views

A problem about direct product

Let $G = U \times V$ where $U, V$ are nonabelian simple groups. Then $G$ has precisely four different normal subgroups. When I was showing the above statement, I did not use the fact that $U, V$ are ...
7
votes
4answers
2k views

Order of nontrivial elements is 2 implies Abelian group

If the order of all nontrivial elements in a group is 2, then the group is Abelian. I know of a proof that is just from calculations (see below). I'm wondering if there is any theory or motivation ...
1
vote
3answers
260 views

Question about rings and modulo multiplication tables

Why is 2=0 in K for part a)? Also I don't understand what part b) is asking you to do - what does it mean by alpha = [X], so M=etc. Could someone please explain the question and the solutions to part ...
2
votes
3answers
608 views

If $H\leq Z (G)$ and $G/H$ is nilpotent, then $G$ is nilpotent.

Let $G$ be a group. If $H\leq Z(G)$ and $G/H$ is nilpotent, then $G$ is nilpotent. To prove this proposition, first, I have tried to show that $G/Z (G)$ has non-trivial center. I can see that all ...
3
votes
2answers
250 views

Why are only the first four alternating groups are non-simple?

I know asking for intuition in math is a generally flawed approach, but can anyone give any reason why only the first four alternating groups are non-simple?
0
votes
0answers
94 views

Intersection of finite index subgroups of $\mathbb{Z}^2$

Suppose you have a subgroup $\{g_0,g_1,\ldots,\}=:G < (\mathrm{GL}_2(\mathbb{Q}),\cdot)$ of invertible 2 by 2 matrices over $\mathbb{Q}$, given by all matrices with determinant in a given subgroup ...
3
votes
1answer
151 views

Quotient of ideal in group ring is isomorphic to abelianization [duplicate]

Let $G$ be a group and $\mathbb Z G$ the group ring over the integers. Let $I$ be the ideal of elements $\sum_{g\in G} n_g g$ with $\sum_{g\in G} n_g = 0$. I am trying to prove that $I/I^2$ is ...
7
votes
2answers
521 views

What is a good book for a second “course” in group theory?

Having studied some group theory in my last term at university, I've found it to be quite interesting, although it's also something I want to improve on (mainly when it comes to proving statements), ...
7
votes
3answers
699 views

Let $H$ be a subgroup of $G$ such that for all $x \in G$, $x^2 \in H$

Letting $H$ be a subgroup of $G$ such that for every $x\in G$, $x^{2}\in H$. So, which of the following is true? a. $H$ is a normal subgroup containing $G'$. b. $H$ is a normal abelian subgroup. c. ...
2
votes
1answer
116 views

Stuck on constructing my semi direct product

I want to work out $H \rtimes Q$, where $H = C_{17}$ and $Q = C_{2}$. What this means is that I want to work out the groups that map $\theta: C_2 \rightarrow Aut(C_{17})$. $Aut(C_{17}) \cong C_{16}$. ...
2
votes
3answers
240 views

Show that $\mathbb{R}/\mathbb{Z}$ is isomorphic to $\{e^{i\theta} : 0 \le \theta \le 2\pi \}$

This question is asking to prove that the quotient group $\mathbb{R}/\mathbb{Z}$ is isomorphic to the group of complex numbers with modulus 1 (under multiplication). It's hard for me to visualize ...
1
vote
1answer
73 views

How do you know what each element maps to in a semi direct product

I have this example in my lecture notes "Calculate the semid direct product, $H \rtimes Q$ where, where $H = C_7, Q = C_3, C_7 = \langle b | b^7 = 1 \rangle$." $Aut(C_7 \cong C_6$. Find all ...