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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.

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50 views

How big must the union of a group's Sylow p-subgroups be?

For various orders $n$ it's a common exercise to prove that a finite group $G$ of order $n$ can't be simple by using the Sylow theorems to show that there is some prime $p \mid n$ such that the number ...
2
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1answer
20 views

What happens to the group structure of an elliptic curve over a field when the discriminant = 0?

Working on a question for a number theory class. So, basically, it asks us what happens to the group structure of an elliptic curve over a field if the discriminant is equal to zero? So, basically,...
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0answers
11 views

Projection operator in group representation

Assume that we construct a representation for a group $G$ which is reducible. Then to block diagonlize it (decompose it to irreducible one), we first calculate the frequency of each irreps using (here ...
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0answers
28 views

proving that a group of order 60 is simple using homomorphisms.

I want to show that the group G, where $|G|=60$ and has 20 elements of order 3 is simple. Here's what I did: Suppose that G is not simple this implies that $n_3>1, n_2>1, n_5>1$ where these ...
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0answers
16 views

Show that S5 is isomorphic to Aut(A5)

There should be a proof using that there is only one conjugation-class of subgroups of A5 of index 5. This has to be proven also. I simply don't have even a slightly idea of what to do...
0
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0answers
13 views

Group created by Gamma matrices

Suppose we have $D$ gamma marices $$\gamma_{a}\gamma_{b}+\gamma_{b}\gamma_{a}=2\delta_{ab}I$$ with $a,b=1,2,...,D$. $$\ \Gamma^{A} = \{I, \gamma_{a_{1}}, \gamma_{a_{1}}\gamma_{a_{2}}, \gamma_{a_{1}}...
3
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1answer
27 views

lLet $R$ be an integral domain and $P$ be a prime ideal in $R[x]$ such that $P\cap R=\{0\}$. Show that $\text{height } P\le 1$.

Let $R$ be an integral domain and $P$ be a prime ideal in $R[x]$ such that $P \cap R=\{0\}$. Show that $\:\operatorname{height}P\le 1$. please help to find its height...
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2answers
28 views

Units in $R[X]$, where $R=\Bbb Z/p^2q\Bbb Z$ [on hold]

Let $p$ and $q$ be two prime numbers and let $R=\Bbb Z/p^2q\Bbb Z$. find units in $R[X]$. i am not getting how to do such type of problems....
0
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1answer
34 views

Existence of a normal subgroup with finite index

i need to prove that if G is a group and K is a subgroup of G such that |[G:K]|=n>1 then there exists a subgroup N of K such that N is a normal subgroup with respect to G and [G:N]<(n!)+1. I did ...
3
votes
3answers
92 views

From a problem in group theory

When I read in Group Theory of Scott. It has a question, I think it's hard. I have tried to solve it, but I can't. Problem: "If G is a group whose order is odd and less than 1000, then G is solvable" ...
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0answers
30 views

How to find a homomorphic map in following question?

Let $S1$ and $S2$ be two sets. Suppose that there exists a one-to-one mapping $J$ of $S1$ into $S2$ . Show that there exists an isomorphism of $A(S1)$ into $A(S2)$, where $A(S)$ means the set of ...
0
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1answer
12 views

a question about the definition of the Chabauty topology for discrete groups

Let $G$ be a discrete group and let $S(G)$ the set of subgroups equipped with the following topology: A net $(H_i)_i\subset G$ of subgroups converges in the Chabauty topology to a subgroup $H$ of $G$ ...
1
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2answers
47 views

Let $a$ and $b$ be elements of a group $G$, and $H$ and $K$ be subgroups of $G$. If $aH=bK$, prove that $H=K$.

Let $a$ and $b$ be elements of a group $G$ and $H$ and $K$ be subgroups of $G$. If $aH=bK$, prove that $H=K$. My attempt: we know that $aH=bH$ if $a\in bH$ or $a^{-1}b\in H$. I think that these ...
1
vote
1answer
24 views

Quotient group of $C^*$ by Circle group is isomorphic to $R^+$

$C^*$ is the set of complex numbers except $0$ and $S'$ is the circle group. I have to show the quotient group $C^*/S'$ is isomorphic to $R^+$. Let's define $f:C^*→ R^+$, $z→|z|$. Why is $Kerf$ $S'$? ...
3
votes
2answers
70 views

Alternative view on the group of units of $\mathbb{Z}/p\mathbb{Z}$

The multiplication graphs for $\mathbb{Z}_p := \mathbb{Z}/p\mathbb{Z}$ (with $p$ prime) visually reveal some basic facts, especially that each non-zero element has an inverse, or – equivalently &...
1
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1answer
29 views

Any group can be represented as the fundamental group of a 2-dimensional topological space

I saw in the textbook affirmation that any group can be represented as the fundamental group of a $2$-dimensional topological space. Without proof. May be you can give some ideas, how can I proof that ...
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0answers
44 views

Hall subgroup that has normal complement

Let $H$ be a hall subgroup of finite group $G$. If $H\leq Z(N_{G}(H))$ then $H$ has a normal complement in $G$. I want to know if we assume $|H|=n$ , $H$ is the only subgroup of $ N_{G}(H)$ with ...
2
votes
1answer
40 views

If $G$ is a finite group of odd order, show there is only one homomorphism between $G$ and $\mathbb{R}^{\times}$ which is the trivial one.

If $G$ is a finite group of odd order, show there is only one homomorphism between $G$ and $\mathbb{R}^{\times}$ which is the trivial one. I know there is a similar case if the order of G and H is ...
1
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1answer
21 views

Number of homomorphism between a free group $F(S)$ and a group $G$

Let $F(S)$ be a free group with finite rank, $G$ a group with order $n$. I want to know if the universal property can help determine the number of homomorphisms $F(S)\rightarrow G$. Say $H$ is the ...
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0answers
15 views

What is the subgroup of $O_2$ generated by two reflections in $\Bbb R^2$ about two lines of angle $\frac{2\pi}{n}$?

Question from a Math Exam: What is the subgroup of $O_2$ generated by two reflections in $\Bbb R^2$ about two lines of angle $\frac{2\pi}{n}$? I am unable to understand the meaning of this question? ...
2
votes
1answer
66 views

Decompose rational numbers into sum of rational numbers of the form $\frac{1}{a}$.

I need to show that Given $n>0$ and a rational number $q$ there are only finitely many n-tuples $(c_1,...,c_n)$ of natural numbers such that $q=1/c_1+...+1/c_n$. This result can be used to show ...
2
votes
2answers
26 views

Definition of the tensor product of representations

I'm a bit confused about the following definition: Let $\rho_1:G \to Aut(V_1)$, $\rho: G \to Aut(V_2) $ be two representations of the same group $G$. Then a tensor product of representations is ...
2
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2answers
35 views

Longest element of Weyl Group for $G_2$

Let $\mathfrak{g}$ be a semisimple Lie Algebra, $\mathfrak{t}$ a Cartan Subalgebra, $\Phi$ the corresponding set of roots, $\Delta \subset \Phi$ a root basis and $W$ the Weyl Group with respect to $\...
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2answers
35 views

Why is $\phi_1: Z_{16} \rightarrow Z_{16}$ a homomorphism but $\phi_2: Z_{16} \rightarrow Z_{24}$ is not?

Let $\phi_1(x)=4x$ mod $16$ Let $\phi_2(x)=4x$ mod $24$ Even though $\phi_1(a+b) = 4(a+b)$ mod $16$ = $4a + 4b$ mod $16 = \phi_1(a) + \phi_1(b)$ it is not true that $\phi_2(a+b) = 4(a+b)$ mod $24$ = $...
3
votes
2answers
35 views

Identify a group $G$ that has subgroups isomorphic to $\mathbb{Z}_n$ for all positive integer $n$

Identify a group $G$ that has subgroups isomorphic to $\mathbb{Z}_n$ for all positive integer $n$. Source: Gallian Contemporary Algebra, Isomorphism Chapter I am stuck at the very premise of this ...
1
vote
1answer
21 views

Word norm in compact subsets of finitely generated groups

Let $G$ be a finitely generated topological group, not necessarily discrete. Fix a finite generating set $S$ and denote by $|x|$ the word norm of $x \in G$ with respect to this generating set, i.e., ...
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2answers
67 views

Is there always a way to break a group?

Let $G$ be a finite group ( some finite representation is given ) and Let $N$ be a normal subgroup of group $G$. I know that given a $N$ and $G/N$ one can't get back a $G$ ( always ). My question is ...
0
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0answers
37 views

Intersection of normal subgroups is normal [duplicate]

Can someone tell me if this proof is sufficient given the assumption. Let H,K be normal subgroups of G. Assumption: g(H∩K)= (gH)∩(gK) Proof: g(H∩K) =(gH)∩(gK) =(Hg)∩(Kg) =(H∩K)g ...
1
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1answer
27 views

Are there such things as cyclically presented (arrows-only) categories?

Motivation: Let $w$ be an element of the free group $F_n$ on the generators $\{x_i\}_{i=0}^{n-1}$. Define a function $\theta: F_n\to F_n$ by $x_i\mapsto x_{i+1}$ (and extend this to all elements of $...
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0answers
66 views

Is there a categorification of “(virtually) solvable”?

If this question doesn't make sense or is otherwise poor quality, then I'm sorry. Motivation: As part of my research, I study virtually solvable (1) groups. These are goups that have a solvable ...
0
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0answers
44 views

Coloring of the edge of the 1*3 grid

I am trying to find the number of distinct coloring in the following problem: Consider a 1*3 grid (shown as below) using 10 sticks and 8 balls. Color sticks with $m$ colors. How many ways are there ...
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0answers
34 views

The category of special unitary groups

Is there such a thing as the category of special unitary groups and continous group homomorphisms? There is a category of abelian groups and This category factorizes the multi set monad. I am ...
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1answer
37 views

Understanding semidirect product for group of order 30

I am using Dummit and Foote (pg 181-182) and trying to understand section 5.5 on semidirect product for a group of order 30. They start out with a reminder that if $H$ is of order 15 it is a normal ...
2
votes
2answers
33 views

Conjugacy class of permutatuion

I have permutation $\partial$ = (1 10)(2 6 3 7 6 8 12)(4)(9 11) $\in S_{12}$ and i need to find number of elements in conjugacy class of permutation $\partial$ in group of all permutations. I dont ...
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0answers
9 views

Show $C \subset G$ is relatively convex iff $G/C$ is left-orderable.

Prove: Suppose that $C$ is a subgroup of $G$, denote the set of left cosets $\{gC\}_{g\in G}$ by $G/C$. The subgroup $C$ is relatively convex in $G$ if and only if there exists an ordering $\prec$ of ...
1
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1answer
30 views

Show the abelianization of braid broup $B_n$,$n\geq 2$ is isomorphic to $\mathbb{Z}$

I need to show that the abelianization of braid group $B_n$, for $n\geq2$ is isomorphic to $\mathbb{Z}$, and that the commutator subgroup $[B_n,B_n]$ is exactly the set of braids represented by words ...
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0answers
28 views

Relationship between Frattini subgroup and the first Agemo subgroup of a prime-power order group

Let $G$ be a $p$-group for an odd prime $p$. The Frattini subgorup $\Phi(G)$ is defined as the intersection of all maximal subgroups or, equivalently, as the subgroup of non-generating elements, i.e. ...
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3answers
19 views

Prove if $M_{g_1}^A \cap M_{g_2}^A \neq 0 $ then $M_{g_1}^A = M_{g_2}^A $

Let $G$ be a group and $A$ is a subgroup of $G$. Let define $M_g^A := \{ a * g | a \in A \}$ for an arbitrary element $g \in G$. How can i prove that, for any $g_1,g_2 \in G$, if $M_{g_1}^A \cap M_{...
0
votes
3answers
36 views

Show that the set $H = \{2^n:𝑛 \in\mathbb{Z}\}$ is a subgroup of for $\mathbb{Q}\setminus\{0\}$

Do we just use the subgroup criteria on this? Finding that it's closed, has inverse and identity within the subgroup? But I still don't how does that prove the fact that's $H = \{2^n:𝑛 \in\mathbb{Z}\...
2
votes
1answer
44 views

Help constructing a counterexample subset of a non-abelian group

I am asked whether or not there exists an $r$ such that given any subset $A$ of any group (abelian or not) then: $$|A\cdot A| \leq K|A| \Rightarrow |A\cdot A\cdot A| \leq K^r |A|$$ (where $|A \cdot ...
1
vote
1answer
26 views

Is it true that if all elements of a group (multiplication modulo $n$) are composed with any one element, then they become another group?

I read the following question in Gallian's Contemporary Abstract Algebra: Show that the set $A = \{5, 15, 25, 35\}$ is a group under multiplication modulo 40. What is the identity element of this ...
0
votes
1answer
42 views

How to show the following about the infinite group?

Prove, or disprove the statement: An infinite group $G$ can never act transitively on a finite set $X$. What i know is that a group action $G \times X \to X$ is transitive if it possesses only a ...
0
votes
1answer
23 views

An infinite left word in some alphabet

What is the infinite left word in some alphabet? I can understand the definition of the infinite right word -- some path in Cayley graph from $id$-element to $+\infty$ or $-\infty$. But is the ...
0
votes
1answer
45 views

Simple groups with normal generating sets

Is it known that which finite non-abelian simple groups have a minimal generating set $S$ which is normal in $G$ (i.e. for every $g\in G$, we have $gSg^{-1}=S$)?
2
votes
1answer
57 views

The (un)decidability of the Tits Alternative for any given (suitably defined) set of groups.

Please forgive me if this question is ill-formed. I don't know much about decidability. Some Background: There are problems in combinatorial group theory that are undecidable, such as the word ...
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1answer
44 views

What does $\vee$ mean in the context of group theory?

What does $\vee$ mean in the context of group theory? In particular, I am trying to figure understand:
2
votes
1answer
32 views

Minimum number of conjugacy classes of a finite non-abelian group

I am trying to answer the following: Let $G$ be a finite non-abelian group. Show that $k(G) > |Z(G)| + 1$, where $k(G) $ is the number of conjugacy classes of $G$. I can see that each element in $...
0
votes
1answer
30 views

Finding all elements in $GL_{2}(F)$ that commute with all other elements in $GL_{2}(F)$ [duplicate]

I'm new to group theory. I'm trying to find which elements of $GL_{2}(F)$ commute with all other elements of $GL_{2}(F)$. I have proved that all diagonal matrices as following $\begin{pmatrix}a & ...
2
votes
1answer
29 views

Minimum size of index of a proper subgroup of a finite, non-abelian simple group $G$

Let $G$ be a finite non-abelian simple group and $p$ the largest prime divisor of $|G|$. Show that if $H < G $ then $|G : H | \geq p $. This is from a chapter of a book about group actions, ...
0
votes
1answer
22 views

Homomorphism between groups and group order

Let $φ: G \to \mathbb{Z_{15}}$ be a group Homomorphism and $ord(G) = n$. Are the following true or false? 15|n n|15 All I know is that for $a\in G, ord(φ(a))|ord(a)$, but I can't seem to be getting ...