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 studies groups.

learn more… | top users | synonyms (2)

3
votes
1answer
44 views

Is it enough to determine if two finite groups are isomorphic if we can map the generators?

Heading: A General Inquiry about Finite Isomorphic Groups and Their Generators If I am given two finite groups whose generators I know, is it enough to show that by mapping the generators ...
0
votes
0answers
34 views

a representation of permutation groups

The smallest degree of faithful permutation representations of $S_k$ for $k\geq 6$ have degrees $k$ (natural action), $2k$ (imprimitive action on cosets of $A_{k−1} )$, and $k(k − 1)/2$ (action on ...
4
votes
1answer
53 views

Automorphisms of a group and subgroup

Is there a finite $p$-group $G$ such that $G$ has less number of automorphisms than some subgroup: $$|\mbox{Aut}(G)|<|\mbox{Aut}(H)| \mbox{ for some }H\leq G.$$ If there is such a group, then can ...
0
votes
2answers
31 views

Is there a cyclic group such that is isomorphic to Z∗16? [closed]

How do I find an additive group modulo $n$ $Z_n$ which is isomorphic to $Z^*_{16}$ (group with multiplication modulo 16)?
2
votes
1answer
45 views

Automorphism group from non-abelian simple group

Denote $\varphi_G: G \rightarrow Aut G, a \mapsto (x\mapsto axa^{-1})$. I have shown that if $\varphi_G$ is injective, then we have $\varphi_{Aut G}$ is injective. I am now asked to prove that for ...
0
votes
1answer
38 views

Is the dihedral group Dn linearly primitive for n>2?

Let $D_n$ be the Dihedral group (of order $2n$). For $p>2$ a prime number, $\mathbb{Z}/2$ is a core-free maximal subgroup of $D_p$, then $D_p$ is a primitive permutation group, and so linearly ...
2
votes
1answer
56 views

Can the possible groups by determined by hand?

Suppose, $G$ is a group of order $12$ containing an element $a$ with order $4$. Can I show the following facts by hand ? The group is either cyclic or isomorphic to the group $C3:C4$ $a^2$ is the ...
1
vote
1answer
25 views

An inequality for the minimal number of generators of a finite group II

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating exactly the left regular representation (with ...
4
votes
2answers
19 views

Conjugate elements of $SO(3)$ group

Composition of two rotations in 3d space yields another rotation $$R_1 R_2 = R_3, $$ and I can understand this by help of some figures in my book. So, the rotations in 3d space forms group. Then ...
2
votes
1answer
54 views

Quotient $G \to G/N$ induces quotient $H \to H/N$ by restriction?

Let $G$ be a linear algebraic group over an algebraically closed field $k$. Consider closed subgroups $N \subseteq H \subseteq G$ such that $N$ is a normal subgroup of $G$. Then restricting the ...
2
votes
1answer
28 views

If $\alpha,\beta \in S_{n}$, and $\alpha\beta = \beta\alpha$, then $\beta$ permutes those elements left fixed by $\alpha$.

Here is my solution. Let ${a_1,...,a_k}$ represent all the integers that are permuted by $\alpha$, and let ${a_{k+1},...,a_{k+j}}$, where $k+j \leq n$, be all the elements that are left fixed by ...
1
vote
1answer
24 views

Representatons of dimension $1$ on $D_4$

Prove that there are $4$ distinct representations on $D_4$ with dimension $1$ (where the field is $\mathbb{C}$). We have just started learning representations. Getting this question, what ...
0
votes
1answer
27 views

Problem 4 of Section G of Chapter 13 of Pinter's Book of Abstract Algebra

First, some background info/context: Let $G$ be any group of order $10$. Then, by Cauchy's Theorem, there are elements $a, b \in G$ such that $\text{ord}(a)=2$ and $\text{ord}(b)=5$. Since ...
1
vote
0answers
25 views

Baer-Specker group versus free abelian group generated by an uncountable set [duplicate]

I just learned on Wikipedia that the Baer-Specker group, that is, the group of all integer sequences, is not free abelian. I'm hoping I could be helped to understand why this is true by someone ...
1
vote
1answer
44 views

Prove that (possibly) infinite group of all invertible maps of X to itself is not Abelian.

I have this question on my assignment and I this fact seems trivial to me, but I can not come up with a rigorous proof. I thought to go by contradiction: Assume such a group $G$ is Abelian -> ...
2
votes
1answer
25 views

Looking for some insight or explanation in Normal group and similar matrix

I need someone can give me some insight in Normal group and Matrix Similarity. Normal subgroup $g \in G \text{ and } N \lt G$ $gNg^{-1} = N$ Matrix Similarity $B = PAP^{-1}$ It seems to me ...
4
votes
2answers
81 views

Why isn't a set like $\{0,1,3,6,8\}$ a subgroup of $\mathbb{Z}_9$?

Why are there only 3 subgroups of $\mathbb{Z}_9$? What about $\{0,1,3,6,8\}$? There is an identity and inverse for each element in that subset.
0
votes
0answers
45 views

Group Action: Group $(\mathbb{Z}, +)$ acting on $\mathbb{R}$

In my group theory notes I have the following: The Group $(\mathbb{Z}, +)$ acts on $\mathbb{R}$ as follows: $m\in \mathbb{Z}$ and $r\in\mathbb{R}$: $m.x \to (-1)^mr$ in this notation $m.x$ ...
4
votes
1answer
27 views

Group theory commutator and solvable groups

let G be a group such that it contains 2 members $a, b \in G$ that statisfy: $a = p^{-1} b p$ where $p \in G$ $a = q^{-1} [a,b]q $ where $q \in G$ $a,b,[a,b]\neq e$ where $[a,b]$ is the commutator ...
1
vote
0answers
35 views

Can the number of order-sequences of a group of order $n$ be determined efficiently?

Let $n\ge 1$ and $G$ a group of order $n$. Determine the order of every element of $G$ and list the orders increasing. Can the number of possible order-sequences be determined efficiently ? For ...
1
vote
1answer
46 views

Proving isomorphism [duplicate]

I want to prove at any group of order $4$ is isomorphic to either $\Bbb Z_4$ or $\Bbb Z^*_8$. I know that these two groups are not isomorphic to each other because they have different order, but I ...
1
vote
1answer
35 views

representation of symmetric group

Please help me for the following statement. I do not understand what does it mean centralizer of a representation. The imprimitive transitive representation of symmetric group $S_n$ of degree 2n has ...
0
votes
0answers
20 views

Prove that the order of $GL_{2}(\mathbb{{F}_{p}})$ is $p^{4}-p^{3}-p^{2}+p$. [duplicate]

Let $p$ be a prime. Prove that the order of $GL_{2}(\mathbb{{F}_{p}})$ is $p^{4}-p^{3}-p^{2}+p$. I know that the matrix $2\times 2$ is given by this formula ...
0
votes
1answer
27 views

Abelianization of $\mathbb{Z}_2*\mathbb{Z}_3$

Intuitively it has to be $$\text{Ab}(\mathbb{Z}_2*\mathbb{Z}_3)=\mathbb{Z}_2\times\mathbb{Z}_3$$ here is my approach on how to prove it $$\mathbb{Z}_2=P(a\mid a^2),\mathbb{Z}_3=P(b\mid b^3)\Rightarrow ...
0
votes
0answers
26 views

General linear group acting on vector space

I have this question really stuck on it Let G denote general linear two by two matrices over field Fp for a prime p acting on the vector space of two column vectors over Fp i cannot find orbits or ...
0
votes
0answers
46 views

Is it correct? $1^n +2^n +…+(p-1)^n=-1 \pmod p$

$p$ a prime number, $n\in \mathbb{N} $ and $p-1\mid n$ then $1^n +2^n +...+(p-1)^n=-1 \pmod p$ I'm not sure if my proof is correct: Take the group $G=(\mathbb{Z*}_{p},\cdot)$ with the ...
1
vote
1answer
32 views

Determining which elements of a matrix group form a one-parameter subgroup

We have just learned about one-parameter subgroups in my Algebra class and I am not sure if I am approaching the following proof in the right way. Problem Statement: Let $G$ be a group of real ...
0
votes
1answer
31 views

Map from quotient group

I well known that given a group G and a normal subgroup H we can find an epimorphism from G to G/H. I am trying to find conditions both in G and H to find a map in the other directions, it seems ...
0
votes
0answers
67 views

Subgroup of $S_4$ of order 4 [duplicate]

The subgroup of $S_4$ of order $4$ are seven? In fact I have: \begin{align} \langle(12),(34)\rangle = \{1,(12),(34),(12)(34)\} \\ \langle(13),(24)\rangle = \{1,(13),(24),(13)(24)\} \\ ...
2
votes
0answers
26 views

subgroups of order 2 of $S_4$ [duplicate]

The number of subgroups of order 2 of $S_4$ is nine? In fact I have : {1,(12)}, {1,(13)}, {1,(14)}, {1,(23)}, {1,(24)}, {1,(34)}, {1,(12)(34)}, {1,(14)(23)}, {1,(13)(24)}
0
votes
1answer
25 views

Find all the homomorphisms from $C_3$ to $A_4$

Find all the homomorphisms from $C_3$ to $A_4$. I know that $C_3=\{1,c,c^2\}$ and $A_4$ is the group of even permutations on four elements. So let $\lambda:C_3\rightarrow A_4$ be a homomorphism. ...
0
votes
1answer
23 views

Prove an intermediate subset equals one of the groups it lies between

Let $R^*$ be the multiplicative group of nonzero real numbers. Given that $H \leq R^*$ and $R^+ \subseteq H \subseteq R^*$, prove $H = R^+$ or $H = R^*$. My issue is my professor went over this ...
7
votes
2answers
99 views

What is Representation Theory?

I'm beginning a course that uses representation theory, but I do not really understand what that is about. In the text I am following, I have the following definition: A representation of the Lie ...
1
vote
1answer
37 views

linear characters on a Grothendieck ring of a modular category.

I am reading the paper "Rank-Finiteness for Modular Categories" by Bruillard,Ng, Rowell, and Wang. Let $C$ be a modular category and let $K_0(C)$ be the Grothendieck ring generated by simple objects ...
1
vote
1answer
88 views

Factorization of conjugacy equation's solutions in Monoids

01-28 Update: In the first version I was claiming that the authors were not explicitly or implicitly but I was wrong so I change my question [long explaination at the end of the question] Two ...
4
votes
4answers
135 views

Group conjecture

Conjecture: Given a finite group $G$ and a subset $A\subset G$. Then $\{A,A^2,A^3,\dots\}$ is a group iff $\forall n\in \mathbb N: |A^n|=|A^{n+1}|$. Given that the composition between the ...
0
votes
2answers
52 views

Sylow subgroups of $\text{SL}_2(q)$.

Let $p,q$ be primes such that $p$ is a divisor of $|\text{SL}_2(q)|=(q-1)q(q+1)$. Hence $\text{SL}_2(q)$ admits non-trivial Sylow subgroups. I am interested in the isomorphism type of the $p$-Sylow. ...
1
vote
2answers
46 views

When $n$ is odd, $\langle (123),(12…n)\rangle$ generates $A_{n}$

Working with the knowledge that the set of 3-cycles generates $A_{n}$, the basic idea is to express any 3-cycle as a word in $(123)$ and $(12...n)$ when $n$ is odd. Not knowing how to progress, I ...
0
votes
1answer
41 views

If $M$ is normal and $M \cap U = U'$ for some special subgroup $U$, then $M / G'$ is a Hall-subgroup of $G / G'$.

Let $G$ be a finite group and $U \le G$ be a finite group of odd order. Suppose that $N_G(U) = TU$ where $T = \langle t \rangle$ for some involution $t \notin U$. Also suppose $U^g \ne U$ implies $U^g ...
0
votes
1answer
20 views

girth of Cayley graphs on abelian groups

I am trying to prove that if $X(G,S) $ is a Cayley graph where $G$ is abelian and $|S|>2$, then $X(G, S)$ contains a $4$-cycle. I found an example proof at the following link: ...
3
votes
1answer
79 views

Is there a choice homomorphism?

Let $\pi : \mathbb{R} \to \mathbb{R}/ \mathbb{Q}$ be the canonical projection. With the axiom of choice we "know" that there are choice functions $\alpha : \mathbb{R}/ \mathbb{Q} \to \mathbb{R}$ with ...
2
votes
2answers
67 views

Group isomorphisms don't preserve everything?

Consider the following claim: Suppose $G$ and $H$ are groups such that $G \simeq H.$ If $N \triangleleft G$ and $ K \triangleleft H $ are normal subgroups of $G,H$ such that $ N \simeq K $, then ...
0
votes
1answer
48 views

Prove $\mathbb{Z}_2 \times \mathbb{Z}_2$ is not isomorphic to $\mathbb{Z}_4$

In trying to understand why there are two sets of groups of order 4. I know that there exists the Vierergruppe of $\mathbb{Z}_2 \times \mathbb{Z}_2$ and the group $\mathbb{Z}_4$ but I do not ...
1
vote
1answer
26 views

Questions of a completely reducible module

Please help to deal with the tasks of: $1)$ Which cyclic groups are completely reducible as a $\mathbb Z$-modules? $2)$ Which cyclic modules are completely reducible over the ring $\mathbb F[x]$, ...
1
vote
0answers
18 views

Recover the two-sided properties from a right-sided identity axiom and a left-sided inverse axiom [duplicate]

At the end of this page: http://dogschool.tripod.com/housekeeping.html It proves the left sided axioms using the right sided axioms. It then asks you at then end to prove the two-sided axioms using a ...
0
votes
2answers
42 views

Sets of positive integers closed under lcm/gcd?

Is there an exact, workable description of sets of positive integers closed under the lcm or gcd operations? In other words, a set of ideals of Z which is closed under intersections or sums. My ...
1
vote
1answer
36 views

Number of normal subgroups from $F_{2}$ which factor groups are isomorphic to $Q_{8}$

What is the number of normal subgroups of the free group of rank 2 $F_{2}$ whose factor groups are isomorphic to the $Q_{8}$? I have a plan solutions. But I can't get a numerical answer. Minimum ...
8
votes
0answers
220 views

An inequality for the minimal number of generators of a finite group [migrated]

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular ...
4
votes
0answers
69 views

The étale fundamental group as a functor

The usual "topological fundamental group" $\pi_1 (X,x)$ of a pointed topological space $(X,x)$ is functorial in the sense that a pointed continous map $f: (X,x)\rightarrow (Y,y)$ induces a ...
0
votes
1answer
23 views

Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$

Let $H<S_{n}$ and suppose $H$ is not contained in $A_{n}$. Write $H$ as $$H=E\cup O$$ where $E$ and $O$ represent the sets of even and odd elements, respectively. Let ...