Questions tagged [group-theory]

For questions about the study of algebraic structures consisting of a set of elements together with a well-defined binary operation that satisfies three conditions: associativity, identity and invertibility.

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How to find different representation of permutation in product of transpositions?

It is known that product of transposition to write is not unique. it can be written in many ways,there is theorem that stats that all representation of product of transposition for a given cycle are ...
kaushal trada's user avatar
2 votes
1 answer
24 views

$\mathrm{Int}(G)$ char $\mathrm{Aut}(G)$ for $G$ a non-abelian simple group

Let $G$ a non-abelian simple group and let $A=\mathrm{Int}(G)$ and $B=\mathrm{Aut}(G)$. I would like to know the solution to $A$ char $B$. However, I know the following. Let $\phi \in B$. a) $\phi(A) \...
Akasa's user avatar
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0 answers
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Computation of the symmetric number of a finite group [duplicate]

I got a bit curious about the concept of the symmetric number of a finite group, and decided to do some computations with GAP to determine their values for some small finite groups. The symmetric ...
Justin Benfield's user avatar
-2 votes
0 answers
22 views

Examples of finite groups and non-trivial normal subgroups with properties

For each of the following parts, give an example of a non-trivial finite group $G$ and a non-trivial normal subgroup $N\leq G$ satisfying the properties. There is no group homomorphism $G/N\...
Timothy Ho's user avatar
0 votes
1 answer
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$G = \langle C_G(a) \mid a \in Q \setminus \{1\} \rangle.$ Proof by induction on $|G|$

I have a question on a proof found in our lecture book on group theory from Gernot Stroth. I do not understand what here is meant by "Induction on $|G|$", which makes it very hard to grasp ...
Stippinator's user avatar
1 vote
1 answer
60 views

Explicitly finding a finite abelian group from a presentation, $\langle a, b~|~a^n = b^n=1, ab=ba, ab^{-1}ab^{-1}=1\rangle$

I have the following group presentation: $ G= \langle a, b~|~a^n = b^n=1, ab=ba, ab^{-1}ab^{-1}=1\rangle $ It is clear that $G$ is a finite abelian group. I am interested in knowing what exactly $G$ ...
eyp's user avatar
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-1 votes
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Every solvable group of order less than 200 must be of order 60,120,168 or 180

I am trying to prove that every solvable group with order less than 200 must have order 60,120,168 or 180. I thought of taking Burnside’s Theorem and rejecting everyone of the groups of order $p^a q^b$...
Mathematician's user avatar
1 vote
0 answers
65 views

In a finitely presented group, the set of all words which equal 1 in the group is a recursively enumerable set.

I was reading "An introduction to the theory of groups" by Rotman, chapter 12, the word problem. I am stuck in the following theorem, Let $G$ be a finitely presented group with presentation $...
Dwaipayan Sharma's user avatar
1 vote
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26 views

Given an $n\times 2$ table of $x_{ij}\in S_n$ and $a\in S_n$, find if $\exists j_1,\dotso j_n\in\{1,2\}$ s.t. $a = x_{1j_1}* \dotso *x_{nj_n}$

Given an $n \times 2$ table of $x_{ij}\in S_n$ permutations and permutation $a\in S_n$ find if $\exists j_1,\dotso j_n \in \{1,2\}$ s.t. $a = x_{1j_1}* \dotso * x_{nj_n}$. Basically the problem is to ...
H-a-y-K's user avatar
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1 answer
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The number of direct sum of elementary abelian 2-groups

Let $G=(Z_2)^n$, I want to know the number of direct sum of $G$($G=H \oplus K$) or a fine upper bound. For $G=Z_2 \oplus Z_2$, I have calculated that all of its direct sum decomposition is as follows: ...
zeyu hao's user avatar
  • 347
-3 votes
1 answer
40 views

All subgroups of a group of integers coprime with 7 [duplicate]

I am to list all the subgroups of $(\mathbb{Z_7^*}, \cdot)$. Recall that $\mathbb{Z_7^*}$ is a set of numbers s.t. $\gcd(a, 7) = 1$ I've listed all of them: $H_1 = \{ 1 \}, \\ H_2 = \{ 1, 2, 4 \}, \\ ...
Avgustine's user avatar
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0 answers
35 views

Consider the function $\phi(x)=\ln x$ from $\mathbb R^+$ (under mult) to the group $\mathbb R$ (under add) . Prove that $\phi$ is an isomorphism. [duplicate]

Okay so I've proved all of the parts needed for an isomorphism except to show that $\phi(x)$ is onto. How do I go about doing that? I know that you start by assuming that $y \in \mathbb{R^+}$ and that ...
Whole Milk Slammer's user avatar
2 votes
1 answer
65 views

Do all the conjugates of $H\le G$ pairwise intersect in subgroups of the same order?

Let $H\le G$. Do all the conjugates of $H$ pairwise intersect in subgroups of the same order? If not in general, does it hold at least for Sylow $p$-subgroups? The question has a trivial answer when $...
Kan't's user avatar
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1 vote
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A Natural Probability Distribution on the Infinite Symmetric Group

Is there a "natural" probability distribution on the set of bijections from $\mathbb{N}$ to itself? Preferably, I would want a distribution which arises from some combinatorial procedure. ...
Miles Gould's user avatar
2 votes
0 answers
24 views

$T_4/\langle\{b^nab^{-n}\mid n\in\mathbb{Z}\}\rangle$ and the real line with a loop attached to each integer point

Bowditch uses an example in his A Course on Geometric Group Theory, to explain a fact that a subgroup $G\leq F$ need not be freely generated even if $F$ is, but I cannot understand some details of it. ...
一団和気's user avatar
-2 votes
1 answer
63 views

Grothendieck group of the Category of Finitely Generated R-Modules is Isomorphic to $\mathbb{Z}$

I was reading the top answer to this post: Torsion Grothendieck group. In the top answer, it states, "This is roughly because the Grothendieck group of the category of finitely generated modules ...
Rocky.Racoon.'s user avatar
-2 votes
0 answers
31 views

Classification of subgroups of $\mathrm{GL}_2(\mathbb{F}_{\ell})$ [closed]

I am trying to solve this exercise sheet by Aaron Landesman on the classfication of subgroups of $\mathrm{GL}_2(\mathbb{F}_{\ell})$ (here). I have completed most of the exercises, but am stuck on 2 in ...
Yang Awotwi's user avatar
3 votes
0 answers
45 views

Subgroup of braid group $B_3$ isomorphic to itself

Consider the braid group $$B_3=\langle x,y:xyx=yxy\rangle.$$ It has a proper subgroup $N$, defined as follows: $g$ is in $N$ if and only if the sum of all exponents in $$g=\prod u_i^{\varepsilon_i},\ ...
atzlt's user avatar
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0 answers
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For groups $F, G, H$, if $G \not\approx H$, does that mean $F \oplus G \not\approx F \oplus H$? [duplicate]

For context, we were asked to prove in my Abstract Algebra class that $$\mathbb{Z}_3 \oplus \mathbb{Z}_9 \not\approx \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3, \tag{1}$$ where $\mathbb{Z}_n$...
Mailbox's user avatar
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1 vote
0 answers
32 views

Are $\phi^{'}_i,\tau_i$ the coresponding compatible functions for equivalence relations constructed in the directed limit of groups/rings.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz, How to Prove it by: Dan Velleman, and An Invitation to Abstract Algebra by ...
Seth's user avatar
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2 votes
3 answers
116 views

Let $\phi: \mathbb Z_{15} \to \mathbb Z_{15}$ with $\phi(2)=5$. What is $\phi(1)?$

Let $\phi: \mathbb Z_{15} \to \mathbb Z_{15}$ with $\phi(2)=5$. What is $\phi(1)?$ Of course, $\phi(2)= \phi(1+1)=\phi(1)+\phi(1)=2\phi(1)=5$. So, $\phi(1)=\dfrac{5}{2}$ but there is not element $\...
Fuat Ray's user avatar
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-2 votes
0 answers
31 views

$ \langle x, [G, G] \rangle = \{ x^mc | m \in \mathbb{Z}, c \in [G, G] \} $ [closed]

As part of proving that "Let G be nilpotent of class k. For every x $\in$ G, the subgroup H generated by x and $[G, G]$ is a normal subgroup, nilpotent of class $\leq k − 1$.", the lecture ...
Kon-kon's user avatar
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1 answer
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Non-Abelian group of order $21$ subgroups

This is an exam problem from a couple of years ago... I)Let $G$ be a nonabelian group of order $21$. What are the possible orders of elements of $G$? II)Find the number of Sylow $p$–subgroups of G, ...
Aristarchus_'s user avatar
2 votes
0 answers
86 views

Any splitting field $K/F$ containing a radical extension $R_t/F$ is itself a radical extension [duplicate]

Any splitting field $K/F$ containing a radical extension $R_t/F$ is itself a radical extension. Hello :) I am math student working on Galois Theory and I came by an exercise that I cannot solve, I ...
Sergio Balda De Frutos's user avatar
4 votes
1 answer
103 views

The category of abelian groups with quasi-homomorphisms

Let $A$ and $B$ be abelian groups. Say that a map $f: A \to B$ is a quasi-homomorphism if there exists a finite $D \subseteq B$ such that $$\forall a_1, a_2 \in A: f(a_1 + a_2) - f(a_1) - f(a_2) \in D$...
Smiley1000's user avatar
2 votes
1 answer
67 views

When does every element of Sylow $p$-subgroup is also a member of another Sylow $p$-subgroup?

Is it possible that every element of Sylow p-subgroup is also a member of another Sylow $p$-subgroup? Let $H_1$,$H_2$,....$H_n$ denotes different Sylow $p$-subgroup of group G, then for every $ x \in ...
femto's user avatar
  • 371
2 votes
0 answers
19 views

Non-monotileable amenable groups

We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$. In his article Monotileable Amenable Groups, B. Weiss gives lots of examples of amenable ...
Saúl RM's user avatar
  • 3,413
0 votes
2 answers
70 views

If all order p subgroups are conjugate, do we have any constraint on p's power on group $G$'s order?

Background from here, the question is related to if all order 4 subgroups are conjugate, then what property do we have. A natural following up question from that post is: If all order 4 subgroups are ...
femto's user avatar
  • 371
0 votes
1 answer
34 views

Number of conjugacy classes in each coset of a semidirect product is the same.

Let us consider a semidirect product $X=G \rtimes \langle\sigma\rangle$, where $\langle\sigma\rangle$ acts on $G$ via some automorphism. Assume all groups are finite and that $\sigma$ has order $b$. ...
Aron's user avatar
  • 117
0 votes
1 answer
50 views

Axioms of Field

I am currently exploring the fundamental of field theory, especially, in its connection to monoid and group. One way we can describe a field $\mathbb{F}$ is using the following axioms: F1. $(\mathbb{F}...
rp23's user avatar
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1 vote
0 answers
32 views

permutation of the determinant according to the groups $S_{n1}$ and $S_{n2}$?

Reading about alternating linear n maps, I found this alternative definition of a determinant, based on its permutation expression (which iteratively sums the product of all the permutations that can ...
MonkeyDL's user avatar
3 votes
1 answer
168 views

Does this type of short exact sequence always split?

Consider the family of short exact sequences $$ 0 \to \mathbb{Z}^m \to G \to \mathbb{Z}/n \mathbb{Z} \to 0 $$ where $G$ is a finitely generated abelian group. This is known to not always split, as per ...
Paul Cusson's user avatar
  • 2,037
1 vote
2 answers
89 views

$g$ lies in the centre of $G$ iff $\vert\chi(g)\vert=\vert\chi(1)\vert$ for all irreducible characters $\chi$

I have been trying the following exercise (Chapter 3, Exercise 10) from Webb's "A Course in Finite Group Representation Theory". Let $g\in G$. Show that $g$ is in the centre of $G$ if and ...
Manan's user avatar
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0 votes
1 answer
36 views

isomorphic groups that are G-equivariant [closed]

Start with two finite groups $A$ and $B$ and a group isomorphism $f$ between them. Let a finite group $G$ act on both $A$ and $B$. By definition $f$ is $G$-equivariant if $g(f(a))=f(g(a))$. Do I ...
tess's user avatar
  • 3
0 votes
0 answers
18 views

Equivalent definition of Spin group in terms of automorphisms [closed]

Let $\mathrm{Cl}(\mathbb{R}^n)$ denote the (real) Clifford algebra on $\mathbb{R}^n$ with respect to the Euclidean inner product. Let $\mathrm{Cl}^0({\mathbb{R}^n})$ denote the even part of $\mathrm{...
geometricK's user avatar
  • 4,730
0 votes
1 answer
25 views

MAGMA: How to efficiently do coercion of element of HomGrp to element of GrpAuto? [closed]

Suppose I have a finite $p$-group $G$ as GrpPC in MAGMA. The computation of the automorphism group $\mathrm{Aut}(G)$ takes a very long time. Suppose that I also ...
Aericura's user avatar
  • 261
0 votes
0 answers
36 views

$S_n$ is generated by adjacent transpositions. Are there any other generating sets of length $n-1$ consisting only of transpositions?

$S_n$ is generated by adjacent transpositions. Are there any other generating sets of length $n-1$ consisting only of transpositions? The motivation behind this question is that the set of adjacent ...
H-a-y-K's user avatar
  • 681
25 votes
0 answers
365 views
+50

If $f(n)$ is the number of groups of order $n$, then is $f(a)\cdot f(b)\leq f(a\cdot b)$?

Let $f(n)$ be the number of groups of order $n$ up to isomorphism. We want to prove that: $$f(a) \cdot f(b) \leq f(a \cdot b)$$ for all nonnegative integers $a$ and $b$. Our progress: If $a \cdot b \...
Jorge Rael's user avatar
1 vote
1 answer
53 views

How to show the following sequence of group homomorphism to be exact?

Background The following is taken from: Graduate Course In Algebra, A - Volume 1 by: Ioannis Farmakis and Martin Moskowitz ....a short exact sequence of groups is just another way of talking about a ...
Seth's user avatar
  • 3,429
1 vote
0 answers
28 views

How to show the following sequence of group homomorphism is exact?

Background The following is taken from: Graduate Course In Algebra, A - Volume 1 by: Ioannis Farmakis and Martin Moskowitz ....a short exact sequence of groups is just another way of talking about a ...
Seth's user avatar
  • 3,429
-1 votes
2 answers
51 views

$G = \mathbf{Z}\times\mathbf{Z}$ and its normal subgroup $\langle (1,0)\rangle$. How is $G/N$ isomorphic to $\mathbf{Z}$? [duplicate]

I am new to quotient spaces and I came across this problem and it confused me: Given the group $G=\mathbf{Z}\times\mathbf{Z}$ and some normal subgroup $\langle (1,0)\rangle$.. The quotient group $G/N$ ...
baslerbuenzli's user avatar
1 vote
0 answers
25 views

Constructing compatible functions $\phi'_i,\phi'_j$ for equivalence relation $\sim$ for directed limit of directed systems of groups.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz, and How to Prove it by: Dan Velleman. Definition 1: Let $(G_i)_{i\in I}$ be ...
Seth's user avatar
  • 3,429
-1 votes
0 answers
29 views

Let $H=\{A\in G\; |\; \text{det}(A)=3^k, k\in\mathbb Z\}$, $G=GL(2,\mathbb R)$. $H\le G$ . Prove that $H$ is a normal subgroup of $GL(2,\mathbb R)$ [closed]

Not really sure how to solve this one. The full question is: Let $H=\{A\in G\; |\; \text{det}(A)=3^k, k\in\mathbb Z\}$, where $G=GL(2,\mathbb R)$. It is a fact that $H\le G$ (and you don't need to ...
Whole Milk Slammer's user avatar
1 vote
0 answers
69 views

About semi-direct product of two cyclic groups

The following question is related to seeing semi-direct products as subgroups: Let $G$ be a non-nilpotent group and $U = \langle x \rangle$ be a cyclic characteristic subgroup of the Fitting subgroup $...
Siddhartha's user avatar
3 votes
2 answers
214 views

G has a element of order 2 not lying in center

Let G be order of 8,if G has a element of order 2 not lying center,how to prove G is isomorphic to $D_8$? A hint is consider the sylow-2 subgroup of $S_4$, I know it's $D_8$. So I want to construct a ...
ckx's user avatar
  • 416
0 votes
1 answer
48 views
+50

Issue with proof that SU(2) conjugation on Pauli matrices acts via SO(3)

I want to prove that for a given $U\in SU(2)$, $\phi_U(\vec x)=U\vec x \cdot \vec \sigma U^{\dagger}$ can be identified with a $R(U)=R\in SO(3)$ such that $\phi_U(\vec x)=(R\vec x)\cdot \sigma$. ...
user62783's user avatar
  • 127
0 votes
1 answer
37 views

Relation between rotation groups and spin groups

Sorry if very dumb question. What exactly is the relation between rotation groups and spin groups? I've heard that the spin groups are the double cover of rotation groups but ks there a more precise ...
Tomás's user avatar
  • 57
-2 votes
0 answers
21 views

Given a p-group say G of order p^n (n>2); does G have a subgroup H of order p^(n-1) [duplicate]

further can this be continued so if H is the subgroup of order p^(n-1); does H itself then have a subgroup K of order p^(n-2) etc? also H is a subgroup of G; is K also a subgroup of G ? I 'll call K a ...
Gregor's user avatar
  • 1
4 votes
4 answers
670 views

The rotation symmetry group and the reflection group: Is there a name for what they have in common?

When I turn on my monitor, the brand name fills the screen. But since I mounted my monitor upside down so I can watch it in bed looking up, the power-on screen is upside down. But I noticed that it ...
Duce ex Machina's user avatar
-1 votes
0 answers
51 views

Generator of a subgroup of a Galois group [closed]

Let $f(x) = x^4 - 3 \in \mathbb{Q}[x]$. The splitting field of $f(x)$ over $\mathbb{Q}$ is $K = \mathbb{Q}(\sqrt[4]{3}, i)$, where $i^2 = -1$. We are asked to find $[K:\mathbb{Q}]$ first. Defining $E =...
Aristarchus_'s user avatar

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