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)

1
vote
0answers
38 views

Symmetric group $S_4$ [duplicate]

Let $G$ be the Symmetric group $S_4.$ Give a representative of each conjugacy class of $G.$ Then calculate the size of each conjugacy class. I have no idea how to do this.
3
votes
3answers
98 views

Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups

I have a finite $p$-group $G$ and a normal subgroup $N$ which is not the trivial subgroup. I am asked to show that $|N \cap Z(G)| > 1$. There has been a similar question on MSE here: How to show ...
1
vote
0answers
23 views

Prove $N\cap Z(P)\ne e$ given a $p$-group $P$ and a normal subgroup $N$. [duplicate]

Let $P$ be a $p$-group and let $N\triangleleft P$. Prove $N\cap Z(P)\ne e$ . Here's what I know so far: $P$, being a $p$-group, is nilpotent and therefore is solvable. That means that it has its ...
0
votes
1answer
59 views

The order of two subgroups product is not greater than the product of two subgroup orders $|AB|\le |A|\times|B|$

Let $G$ be a group, and let $A,B\le G$ be finite subgroups of $G$. The product $AB$ is defined as following : $AB =\{a\times b | a\in A, b\in B\} $ Show that $|AB|\le |A|\times |B|$. My attempt so ...
6
votes
1answer
131 views

An example of a simple infinite $2$-group

Is there an example of a simple infinite $2$-group? Informations If a $2$-group is Artinian I know that it also locally finite, so the simple $2$-group cannot be Artinian. Take the subgroup ...
1
vote
2answers
82 views

First Order Logic and groups: Prove that no $T$ exists such that $M \vDash T \cup T_{grp}$ **IFF**

I'm learning First Order Logic by myself using a University textbook, and it has the following question in it (as a self exercise): Let $L = \{\cdot, e \}$ the language of Groups and let $T_{grp}$ be ...
0
votes
2answers
34 views

If $G$ is a group and $y \in G$\ $\langle x \rangle$ , then can $\langle y \rangle$ be $G$?

If $G$ is a group and $e \ne x \in G$ and $y \in G$\ $\langle x \rangle$ , then can $\langle y \rangle$ be $G$ ?
2
votes
0answers
52 views

What is the group $G=\mathbb Z_2+\mathbb Z_2$?

I'm trying to figure out what is the group $G=\mathbb Z_2+\mathbb Z_2$ where $\mathbb Z_2 = \{0,1\}$ with the operation addition modulo $2$. I tried to find this group by adding elements from ...
1
vote
0answers
54 views

Simple question on the unitary representation.

Let $\pi_1,\pi_2,\pi_3$ are all irreducible, unitary representation of some algebraic group G. Then is it ture that $$Hom_{G}(\pi_1,\pi_2 \otimes \pi_3) \simeq Hom_{G}(\pi_1 \otimes ...
3
votes
0answers
80 views

Motivation behind automorphism bases?

Given a model $\mathcal{M}$ with a domain $M$ and $B \subseteq M$, $B$ is an automorphism base for $\mathcal{M}$ iff $\forall f \in Aut(\mathcal{M}). (\forall b \in B. f(b)=b) \implies f = ...
2
votes
1answer
66 views

2-transitively, formula [closed]

Let $G$ be a finite group and let $X$ with $|X| \ge 2$ be a set on which $G$ acts. Then $G$ acts on $X \times X$ via $g \cdot (x, y) = (g \cdot x, g \cdot y)$. The action of $G$ on $X$ is called ...
3
votes
1answer
44 views

My proof that $G(x)\to G / G_x$ is injective

Please could someone check my proof that $\varphi : G(x) \to G/G_x$ is injective? The notation is the following: $G$ is a group acting on a set, $G_x = \{g \in G\mid gx = x \}$ and $G(x) = \{gx ...
3
votes
1answer
90 views

let $x$ be in finite group $G$ and let order of $x$ is $p$. If $h^{-1}xh = x^{10}$ for a finite group show that $p=3$ [duplicate]

Let $G$ be a finite group, $p$ be the smallest prime divisor of $|G|$ and x $\in$ G an element of order $p$. Suppose $ h \in G $ is such that $h^{-1}xh = x^{10}$. Show that $p = 3$. I cant ...
1
vote
2answers
145 views

Find the right cosets of $H$ in $G$ simple example

Question: Let $G$ be a group and $H<G$ a subgroup with $|G:H|=2$ Show that the right cosets of $H$ in $G$ are $H$ and $G\backslash H$ Answer given: There are two right cosets, they are disjoint ...
1
vote
0answers
71 views

If two groups $G$ and $H$ both have a nonzero homomorphism to all other groups, and have only two idepotent homemorphisms, are they isomorphic?

Consider to Groups $G$ and $H$ such that: The only two endo-hom omorphisms (homomorphisms from a group to itself) that are idempotent (any function $f$ such that $f(f(x))=f(x)$ for all x) are the ...
2
votes
1answer
88 views

Proving that a subgroup $|H|=p^k$ is a Sylow subgroup of $|G|=p^km$, $m\nmid p$

I'm attempting to prove Sylow's theorems following the sketch described in the Wikipedia article, but I've run into a little hitch since the theorems are presented in a few slightly different forms in ...
2
votes
0answers
62 views

Number of permutations of order m

Is there a closed form for the number of permutations (on n letters) that have order m? If not, is there a tight upper bound?
1
vote
2answers
80 views

Find the orders of $3 + \langle 6 \rangle$ and $2 + \langle 6 \rangle$ in $\mathbb{Z}_{15}/\langle 6 \rangle$

Find the orders of $3 + \langle 6 \rangle$ and $2 + \langle 6 \rangle$ in $\mathbb{Z}_{15}/\langle 6 \rangle$. To what group is $\mathbb{Z}_{15}/\langle 6 \rangle$ isomorphic? I know that: ...
5
votes
1answer
67 views

Any infinite property (T) subgroup of $Aut(F_n)$?

I heard that it is still an open problem whether $Aut(F_n), n\geq 4$ has Kazhdan's property (T), where $F_n$ denotes the non-abelian free group on $n$ generators, my question is: Does there exist any ...
2
votes
1answer
89 views

Factor Group Lemma of Cayley Graph

Factor Group Lemma: Suppose that 1.$N$ is a cyclic, normal subgroup of group $G$. 2.$(s_1,s_2,\ldots,s_m)$ is a hamiltonian cycle in $Cay(G/N;S). 3.The product $s_1s_2\cdots s_m$ generates $N$. ...
2
votes
1answer
33 views
1
vote
1answer
37 views

How to determine if the subset $K=\{ g\in S_4|2^g=2\}$ is a subgroup of $S_4$?

I'm not sure exactly how to start. I know that the group $S_4$ is a finite set of 4 symbols whose elements are all the permutations of the 4 symbols which sums up to 24 permutations. That is, $S_4$ ...
1
vote
1answer
131 views

Group theory- rank of a group. What am I doing wrong?

I was given a question: Let $n\in \mathbb{N}$ and let $A$ and $B$ groups, both isomorphic to $\mathbb{Z}^n$. Let $f:A \to B$ be a surjective homomorphism. Prove $f$ is an isomorphism. Here's my ...
1
vote
1answer
39 views

Understanding transformation as algebraic structure

I am confused about the following structure, and would be very thankful if somebody could give me a hint. Let $\mathbb{S}$ be a set with n elements $\mathbb{S}=\{a_1, a_2, ..., a_n\}$, and $(x,y) \in ...
0
votes
1answer
197 views

Simple confusion: How can rotation matrix be associative but not abelian?

On the Wikipedia page it is said that the rotation matrix is associative but it also states that rotation matrices are not abelian. The associative property (I think) implies that we have the ...
1
vote
4answers
104 views

What is the order of the alternating group $A_4$?

When I write out all the elements of $S_4$, I count only 11 transpositions. But in my text, the order of $A_4$ is $12$. What am I missing? ...
1
vote
1answer
74 views

Discrete quotient group

I have a hard time understanding quotient groups. For example, I need to make sense of the expression $$\mathcal{S}_3 (1,3,5) / \mathcal{Z}_2 (3,5).$$ Here, $\mathcal{S}_3 (1,3,5)$ is a symmetric ...
3
votes
2answers
78 views

The group $\langle a,b,c \ | a^3,b^2, ab=ba^2, c^2, ac=ca, bc=cb \rangle$?

The group $\langle a,b,c \ | a^3,b^2, ab=ba^2, c^2, ac=ca, bc=cb \rangle$ is isomorphic to which permutation group. I have calculated its order and it is $12$, so my guess was $A_4$ but it is not ...
2
votes
0answers
52 views

Spin group Spin(4,1)

i'm interested in the spin group $Spin(4,1)$ wich correspond to the symplectic group $Sp(1,1)$. The only source that I could find about it was wikipedia (http://en.wikipedia.org/wiki/Spin_group). It ...
2
votes
0answers
36 views

Adding relators and normal closure

I learned the notion of group presentation. By definition, a group described by a presentation is the quotient of some free group by the normal closure of the relators. In general, given a group ...
1
vote
2answers
56 views

How do I prove $N(aHa^{-1})= aN(H)a^{-1}$ [closed]

Here $a \in G$ and $H$ is a subgroup of $G$. I have no idea where to begin with this. Please help?
1
vote
0answers
63 views

Show that $S^2=G$. [duplicate]

I try to solve a question from Grove's Algebra book. The question is that: Suppose $S$ is a subset of a finite group $G$, with $|S| > \dfrac{|G|}{2}$. If $S^2$ is defined to be $\{xy:x, y \in S\}$ ...
0
votes
0answers
100 views

On the contragredient representation

Let $\pi$ be a representation of group $G$.Then its contragredient representation $\pi^{\vee}$ is defined by $\pi^{\vee}(g)=^{t}\pi(g^{-1})$. (here $^t$ means the transpose) But I heard that it is ...
5
votes
0answers
62 views

Graphing elliptical curves based on group operation

I just found this and it blew my mind (he gives an elliptical curve to do multiplication). If I understand correctly (from reading the link and other things) the Abelian group he is using is ...
0
votes
1answer
172 views

Properties of $Hom_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Q} )$

I have to prove that : 1) $Hom_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Q}) \cong \mathbb{Q}$ as abelian groups 2) $End_{\mathbb{Z}}(\mathbb{Q}) \cong \mathbb{Q}$ as rings What I have done: 1) We can ...
0
votes
0answers
81 views

Relationship of irreducible polynomial of prime degree $p$ and the full symmetric group $S_p$

According to Wikipedia, if $f(x)$ is an irreducible polynomial of prime degree $p$ over $\mathbb{Q}$ and it has two nonreal roots, then the Galois group of $f$ is the full symmetric group $S_p$. For ...
2
votes
0answers
71 views

Are there any applications for complex analysis in population dynamics?

Strange question: could complex analysis be used to understand population dynamics? I'm interested in modelling dominance hierarchies, mating relationships, and illness behaviour in ancient ...
0
votes
2answers
112 views

Let $G$ be a finite group and $S$ set of generators. Show that every element of $G$ can be written in form $x_1…x_n$ where $x_i\in S$.

I know what does set of generators mean and that subset of $G$ generates some subgroup. But I have no idea how to prove the statement above in title. It's like comes from definition and there is ...
1
vote
1answer
42 views

Question about a finite Galois extension over a field of characteristic $0$

I have a question from an old algebra prelim in Galois theory and would like to know the best or quickest method in proving it. The question is that suppose $F$ is a field of characteristic $0$. ...
1
vote
0answers
48 views

Show that the elements $x^iy^j$ such that $i=0,1,2,3$ and $j=0,1$ are distinct elements of $G$, and hence constitute all elements of $G$.

There exists a group $G$ of order $8$ having two generators $x,y:x^4=y^2=e$ and $xy=yx^3$. I found that $G=\{e,y,x,xy,x^2,x^2y,x^3,x^3y\}$ But how to show that these elements are distinct? Is saying ...
0
votes
1answer
94 views

Group of order $6435$ is not simple

I'm trying to use the congruence relations from Sylow's Theorem to show that a group $G$ of order $|G|= 6435= 3^2 \cdot 5 \cdot 11 \cdot 13$ is not simple. To work with the least amount of Sylow ...
3
votes
1answer
109 views

Why is the order of the subgroup 3?

I want to find the order of the subgroup $\langle ab\rangle$ of $D_3=\langle a,b\mid a^3=1,b^2=1,ba=a^2b\rangle$ According to my notes, the order of this subgroup is 3. But why is it like that? I ...
2
votes
1answer
36 views

Given group $G$, $N \triangleleft G$, $H < G$ prove that $G = N\rtimes H$ iff $f:H\rightarrow G/N, f(h)=hN$ is an isomorphism.

Given group $G$, $N \triangleleft G$, $H < G$ prove that $G = N\rtimes H$ iff $f:H\rightarrow G/N, f(h)=hN$ is an isomorphism. If I assume that $G=N\rtimes H$ then I've proved that $f$ is a ...
2
votes
1answer
32 views

Prove that $G/N \simeq GL_n(F)$

Given field $F$, define $T_{A,w}:F^n\to F^n$ by $T_{A,w}(v)=Av+w$ where $A\in M_n(F)$ and $w\in F^n$. Define $G=\{T_{A,w}|A\in GL_n(F),w\in F^n\}$ and $N=\{T_{I_n,w}|w\in F^n\}$. The first two parts ...
2
votes
1answer
40 views

Symmetric Group acting on $X \times X$

The symmetric group $S_n$ acts on the set $X = \{1,\ldots,n\}$ and hence acts on $X \times X$ by $g(x,y) = (gx, gy)$. Determine the orbits of $S_n$ on $X \times X$. Not sure how do I actually ...
3
votes
2answers
169 views

Do all polynomials of degree n with indeterminate coefficients have Galois groups that are isomorphic to Sn?

I just finished reading "A Book of Abstract Algebra" by Charles C. Pinter and, as someone who is studying this independently, I was having some understanding issues and many questions. 1) Does every ...
0
votes
3answers
103 views

True or not: If a normal subgroup and its quotient are commutative, then the group is.

Let $G,*$ be a group and $A,*$ a normal subgroup of $G,*$: $$ A \triangleleft G \quad( \equiv \forall g\in G: gA = Ag) $$ Then $G$ is commutative iff A and $\frac{G}{A}$ are commutative. I can see ...
4
votes
1answer
158 views

Show that $G/H\cong S_3$

If $G:=S_4$ and $H:=\{id,(12)(34),(13)(24),(14)(23)\}$ Show that $G/H$ has order $6$ and all of its elements have order less than or equal to $3$ (so by the classification of the groups of order 6 ...
8
votes
3answers
581 views

Extension of a group homomorphism

Let $G$ and $K$ be (possibly non-Abelian) groups and let $\phi:G\rightarrow K$ be a homomorphism. Let $\bar{G}$ be a group containing $G$ as a subgroup. Is it always possible to extend $\phi$ to a ...
2
votes
2answers
31 views

Extension of group homomorphsims [duplicate]

Let $G$, $G'$ be groups, and $H$ a subgroup of $G$. Given a homomorphism $f:H\longrightarrow G'$, does there exist a homomorphism $\tilde{f}:G\longrightarrow G'$ such that $\tilde{f}|_{H}=f$?