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
30 views

Looking for group $G$ such that $G$ nilpotent, $N \leq G$, $N \cap Z(G) = \{1\}$

$G$ nilpotent, $N \leq G$, $N \cap Z(G) = \{1\}$. Note that it is a common exercise to show that if we add the assumption t hat $N \triangleleft G$ then $N \cap Z(G) \neq 1$. So I'm really looking for ...
0
votes
2answers
59 views

For a group whose order is the product of two distinct primes, What is the order of the center of the group?

I was solving some questions on group theory and I came across a problem something like this: Let $G$ be a group of order 77. Then the order of the centre of the group is: My attempt: $77= 11\...
4
votes
1answer
53 views

Similarity between Axiomatic Set Theory and Modern Algebra

Continuum Hypothesis(CH) is independent of ZFC Axioms, which means there exist models of ZFC where CH is true, and models where CH is false. Can I say something similar for groups? Something like the ...
1
vote
2answers
54 views

Shoe the group $SL_2(R)$ is can be created by commutators [duplicate]

I want to show that $SL_2(R) = [SL_2(R),SL_2(R)]$. I reduced the problem only show that the matrices $A=\{(1,x),(0,1)\}, B=\{(1,0),(x,1)\}, C=\{(x,0),(0,\frac {1}{x})\}$ are products of commutators....
2
votes
0answers
32 views

Minimal generating set for Dih4 x Z2

I'm trying to find the minimal generating set for a square prism under reflection. This group is $\text{Dih}_4\times Z_2$. Geometrically, the set $\{a,b,c\}$ where $a$ is a rotation of $90^\circ$ ...
-4
votes
1answer
34 views

The possible element orders of $S_{7}$. [closed]

Show that the possible orders of the elements of group $S_{7}$ part of the set $\{1,2,3,4,5,6,7,10,12\}$
2
votes
5answers
80 views

Why is $S_3$ not normal in $S_4$?

I read on this site that $S_3$ is not normal in $S_4$, but I have not been able to prove it. Admittedly, I have not tried brute force because there must be a smart approach. I have tried to find a ...
5
votes
2answers
104 views

If $x,y,z \in G$ and $[x,y] = y, [y,z] = z, [z,x] = x$ then $x=y=z = e$ [duplicate]

Let $G$ be a group. Let $x,y,z\in G$ be three elements. I want to prove that if $[x,y] = y, [y,z] = z, [z,x] = x$ (when $[a,b] = aba^{-1}b^{-1}$ is the commutator) then $x=y=z = e$ It looks simple ...
0
votes
1answer
55 views

How to prove that a set is universe of a subgroup?

I'm trying to solve this proposition: Let $ \langle A, \cdot, ^{-1}, 1\rangle $ and $ \langle B, \cdot, ^{-1}, 1\rangle $ be groups and let $ \alpha \colon A \to B $ be a homomorphism. Then the ...
0
votes
0answers
39 views

For which numbers $l \leq |K|$ field $K$ has subfield, which has $l$ elements? [duplicate]

char$(K) = p$, $|K|= p^n$, $|K^{\times}| = p^n-1.$ We know that $K^{\times}$ is cyclic group. Let $H$ be a cyclic subgroup of $K^{\times}$. Subgroup of cyclic group is also cyclic, so $H$ is cyclic ...
3
votes
1answer
76 views

When is the profinite completion of a centerless group itself centerless?

Having a centerless profinite completion leads to some nice properties. For example, given a short exact sequence $$1\to A\to B\to C\to 1$$ where $A$ is finitely generated and $\hat{A}$ has trivial ...
0
votes
0answers
19 views

Subgroups of $SL(n,F)$ that contain $B \cap SL(n,F)$

The exercise is from the book by Alperin 'Groups and Representations' If $B$ is the usual Borel subgroup of $GL(n,F)$. Determine all the the subgroups of $SL(n,F)$ which contain $B\cap SL(n,F)$. So $...
3
votes
2answers
90 views

Why only define word only with finite product of elements?

My textbook says that the definition of a word is $(s_1,s_2, s_3 , \dots ) $ where $ s_i\in S\cup S^{-1} $ and $s_i =1 \; $ for all $ i $ sufficiently large. I don't get that why is this "$s_i =1 \; $ ...
1
vote
2answers
29 views

Abelianization is left adjoint to the forgetful functor

I am trying to show that the map $ab:Grp \rightarrow AbGrp$ is left adjoint to the forgetful functor. I have seen in a similar question the approach is as follows: Let $f:G\rightarrow H$ be a group ...
2
votes
1answer
55 views

Intersection of Normal closure and Center

Let $G$ be an HNN-extension $\langle a,t\mid t^{-1}a^2t=a^2\rangle $. Then I am going to show that the normal closure of $t$ intersects the center of $G$ trivially. In the first step by using normal ...
4
votes
1answer
141 views

Does there exist some sort of classification of horny groups?

It is well known, that any finite group of order $n$ is isomorphic to a subgroup of $S_n$. Let’s call a finite group $G$ horny iff it is not isomorphic to any subgroup of $S_{|G|-1}$ . Does there ...
1
vote
0answers
28 views

Galois descent for a semisimple automorphism

Let $K$ be a perfect field and $\overline{K}$ be the algebraic/separable closure. Let $V$ be a finite dimensional $K$-vector space, and let $V_{\overline{K}} = V \otimes_K \overline{K}$. Given an ...
1
vote
2answers
49 views

Dihedral group generated by $\langle r,s\rangle$ for all $n$

Under wikipedia for Dihedral groups it claims the following: The $2n$ elements in $D_n$ can be written as $\{e,r,r^2,r^3,\ldots,r^{n-1},s,rs,r^2s,\ldots,r^{n-1}s\}$. I know why this is true and it ...
3
votes
2answers
77 views

Proving commutator subgroup is normal

Part of Aluffi II.7.11 suggests proving this claim. It is actually quite straightforward (especially after some earlier problems in the book): I've managed to prove it by noticing that for arbitrary $...
0
votes
0answers
15 views

Conditions to determine if a matrix is a Lorentz matrix and if it is a Galilean matrix

Consider a generic 4x4 matrix $\Lambda $ $${\begin{bmatrix}{\Lambda ^{0}}_{0}&{\Lambda ^{0}}_{1}&{\Lambda ^{0}}_{2}&{\Lambda ^{0}}_{3}\\{\Lambda ^{1}}_{0}&{\Lambda ^{1}}_{1}&{\...
0
votes
1answer
43 views

Number of elements of order 3 in $C_3 \times C_9$

What is the number of elements of order 3 in the internal direct product $C_3 \times C_9$ of $C_3$ and $C_9$ where $C_i$ is the cyclic group of order $i$. My work so far, let $(a,b) \in C_3 \times ...
1
vote
0answers
99 views

Burnside's Lemma and the inclusion-exclusion principle.

Let $G=\lbrace g_i,i=1,\dots,|G|\rbrace$ be a finite group acting on a finite set $S$. Denote by $\mathcal{O}$ the set of orbits of $S$ induced by the action. Now, by the inclusion-exclusion ...
3
votes
1answer
53 views

Counterexample in permutations of $S_A$ with A an infinite set

I have been going through Pinter's A Book of Abstract Algebra recently and one question bugs me more than any other. When discussing the properties of permutations on a general set $A$, he asks ...
2
votes
0answers
47 views

Does there exist a nontrivial abelian group $A$ such that $\mathrm{Aut}(A) \cong A$? [duplicate]

Does there exist a nontrivial abelian group $A$, such that $\mathrm{Aut}(A) \cong A$? Here $\mathrm{Aut}$ stands for the automorphism group. What do I currently know: If such $A$ exists, it has ...
2
votes
2answers
62 views

Getting a one to one morphism $p : G \to GL_n(\mathbb{C})$

I am studying representation theory, and I would like to find an algorithm that finds, given a finite group $G$, a one to one morphism $p : G \to GL_n(\mathbb{C})$ (the integer $n$ is also found by ...
1
vote
1answer
51 views

Irreducible lattices in $G=G_1\times G_2$

First, we shall recall the definition of an irreducible lattice. Let $G$ be a Lie group which admits a direct product decomposition into simple non-compact factors $G_1\times\dots\times G_k$. A ...
3
votes
1answer
36 views

Representation Theory Block Diagonalizing

I am currently examining the symmetric group $S_4$, and I was tasked with finding a $2$-D, a $3$-D, and a $4$-D representation of the group. The $4$-D representation is reducible, so I first found it ...
1
vote
2answers
52 views

Proving subgroup generated by elements of a given order is normal

Aluffi II.7.7 suggests proving the following: let $G$ be a group, $m$ a positive integer, and let $H \subseteq G$ be the subgroup generated by all elements of order $m$ in $G$. Prove that $H$ is ...
1
vote
1answer
43 views

Show than $(1,k+1), (1,2,3,…,n)$ generate the group $S_n$ if and only if $k$ and $n$ are coprime

I am able to prove constructively that if $(k,n)=1$ then we can generate $S_n$ but am struggling with the converse.
0
votes
1answer
43 views

discrete subgroup of complex Lie group is normal automatically?

This is in relation to Kodaira's Complex Manifolds and Deformation Complex Structures Chpt 2, Sec 2. $W$ is a complex Lie group. A discrete subgroup $G\leq W$ gives properly discontinuous and fixed ...
0
votes
0answers
29 views

Is the unitary group $U(2) \cong SU(2) \times T$ [duplicate]

Let $U(2)$ denote the group of all invertible $2×2$ complex matrices $A$ with $A \overline{A}^T=I$ where $T$ denotes transpose matrix. Let $SU(2)$ be the subgroup of $U(2)$ consisting of those ...
0
votes
1answer
63 views

In S4, what is the subgroup generated by the cycle (123)?

Let $S_4$ be the symmetric group of degree $4$ and $H$ the subgroup of $S_4$ generated by $(1\ 2\ 3)$. I want to list out the members of $H$. I know they are the powers of (123), but I get (132) ...
0
votes
0answers
7 views

Smallest subgroup of rotations for which certain tensor invariance properties hold

I have been able to do up to (b) (i), however, I am not sure where to begin on (ii). Any help would be much appreciated.
1
vote
0answers
73 views

Let $C_8$ be the cyclic group of order eight. Let $G=(C_8)^{10}$ and $H=(C_8)^3$ and $H$ is a subgroup of $G$.

How many exist subgroups $F$ in $G$ such that $F=(C_8)^5$ and $H \cap F$ is $C_8$? I yet understood that there are $2\cdot 8^{i-1}-4^{i-1}$ ways to choose $C_8$ from $(C_8)^i$. So I think that the ...
4
votes
0answers
77 views

Quantum representation of a system of identical particles

I'm studying mathematics and I began a course in quantum statistics, in which I got to the discussion related to indistinguishibility of particles. My professor's notes are not very clear and ...
0
votes
1answer
64 views

Show that GL(n,R) is isomorphic to GL($R^n$)

I read in my book that the said isomorphism holds, but I am confused as to what exactly GL(Rn) is. Can someone help clarify this and how to show the above isomorphism.
3
votes
2answers
77 views

Roots of $f(x)$.

Let $p$ be a prime not equal to $2$. Let $f(x)$ be an irreducible polynomial over $\mathbb{Q}$ of degree $p$ with Galois group isomorphic to the dihedral group $D_{2p}$. I need to show that $f(x)$ has ...
0
votes
1answer
51 views

Is it true that if two groups have same number of elements of order n, then they are isomorphic? [duplicate]

I know that if two groups are isomorphic, they have same number of elements of order n. But will the converse holds? I think it won't as there are many group characteristicss like abelian or cyclic ...
-3
votes
1answer
47 views

Is F_p isomorphic to F_p^2? [closed]

Reasoning was that F_p and F_p^2 have the same characteristic and all fields with the same characteristic are isomorphic to each other. We can then represent all of these fields as GF(p) ?
0
votes
0answers
34 views

Find $(a,b),(c,d)\in G$ so $(a,b)H=(c,d)H$

I had an exam today and there was the following question: Let $G=\mathbb{Z}\oplus\mathbb{Z}$ be a group and $H=2\mathbb{Z}\oplus3\mathbb{Z}$ its subgroup. Let $(a,b),(c,d)\in G$. Find a condition ...
0
votes
0answers
39 views

Induced representation as a submodule of group algebra

Let $G$ be a finite group and $H$ be a subgroup. Suppose $\rho:H \to \mathbb{C}$ is a one-dimensional representation of $H$. This allows us to define the induced representation $Ind_H^G\rho$. I want ...
0
votes
0answers
58 views

All elements in a group with certain order

I'd like for some help with this question. Let ${ G = \mathbb{Z}_{10} \oplus \mathbb{Z}_{12}}$. What is the maximal order of an element in that group? My answer: $O(G) = l.c.m.[10,12] = 60$. (...
-3
votes
2answers
52 views

Does $S_3 \times \mathbb Z_6$ have a subgroup of order 9.

Does $S_3 \times\mathbb Z_6$ have a subgroup of order $9$? where $S_3$ is the symetric group. Clearly $|S_3| \times |\mathbb Z_6|=36$ and $9$ does divide $36$, so its possible. But i can not figure ...
0
votes
0answers
61 views

Is $\mathbb Z^{\mathbb N}$ a free abelian group? [duplicate]

While looking over the universal coefficient theorem, I was wondering whether we can assume that groups of the form $\mathbb Z^{\mathbb N}$ are free. If we replace $\mathbb Z$ by a field, the answer ...
0
votes
1answer
51 views

Calculate order, subgroups and normal subgroups

Given the group $G= \langle (1,3,6,7,4,2,5),(1,2,3)(4,5,7) \rangle$. How do i calculate the order of the group, the subgroups and normal subgroups? The order is the number of elements. We have the ...
3
votes
1answer
63 views

Is this a typo in a proof regarding action of a permutation group on a set?

Recently, I asked a question about action of a permutation group on a set here. Let me summarize it. Let $\mathrm{S}_{m}$ be the set of all permutations of $\{1,2,\cdots,m\}$. Then $(\mathrm{S}_{m},...
0
votes
1answer
28 views

Examples of coexact sequences of groups which are not exact

Say a sequence $A\overset{f}{\to}B\overset{g}{\to}C$ of group morphisms is coexact if $\operatorname{Coker}f= \operatorname{Im}g$. The first isomorphism theorem ensures that exactness implies ...
1
vote
2answers
52 views

Using group presentation, does $\langle A = a^3, 1 = A^2 \rangle$ relate to $\langle A = a^2, 1 = A^3\rangle$?

Let $G = \langle A = a^3, 1 = A^2 \rangle$ relate to $H = \langle A = a^2, 1 = A^3\rangle$ be two "group presentations" namely for: $$ \begin{matrix} \cdot & A & a & a^2 & 1 & Aa ...
1
vote
0answers
19 views

Overgroups of the stabilizer of an isotropic vector in orthogonal groups

Suppose $V$ is a $2n+2$-dimensional vector space over $\mathbb{F}_q$. Suppose also that there exists an isotropic vector $v \in V$. It is known that the stabilizer in $GO_{2n+2}^{\varepsilon}(q)$ of $...
3
votes
0answers
45 views

Recommended readings on the relationship between Groups and Geometries.

The following is from a blog on the development of group theory: "Möbius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a ...