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)

2
votes
3answers
29 views

Number of orbits with Burnside's lemma

We color a equilateral triangle by coloring each edge with one of $k \geq 1$ colors. Find a formula for the number of orbits under the action of $D_6$, the dihedral group of $6$ elements, on the ...
9
votes
1answer
84 views

Does $\forall v ( T_1 v = 0 \lor T_2 v = 0 \lor \dots \lor T_n v =0 )$ imply $T_1 = 0 \lor T_2 = 0 \lor \dots \lor T_n = 0$?

Let $V$ and $W$ be vector spaces and $T_1$, $T_2$, $\dots$, $T_n$ be linear transformations from $V$ to $W$, such that for every $v$ in $V$, either $T_1 v = 0$, $T_2 v = 0$, $\dots$ or $T_n v = 0$. ...
1
vote
3answers
690 views

Prove that $G$ is cyclic if $|G|=15$ and $G$ has only one subgroup each of orders $3$ and $5$

Question: Let $\left | G \right |=15$. If G has only one subgroup of order 3 and only one subgroup of order 5, prove that G is cyclic. Looking for useful hints to the above question. Thanks in ...
0
votes
1answer
42 views

Prove that g=e for a finite group G

Suppose G is a finite group of order n and m is relatively prime to n. If $g \in G$ and $g^{m}=e$, prove that $g=e$. Let $\left | G \right |=n$ and $gcd\left ( n,m \right )=1$ Recall: $\left | ...
3
votes
1answer
34 views

For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?

For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$ i.e. $[-1] \ne [2^k] , \forall k >0$ in $\mathbb Z_n$ ? Is there any ( at least sufficient ) ...
-1
votes
0answers
34 views

The order of the group $U(n)$ is even for $n\gt2$ [closed]

Use the corollary to Lagrange's theorem that the order of an element in a group $G$ divides the order of the group $G$ to prove that the order of $U\left ( n \right )$ is even when $n\gt2.$ I ...
1
vote
1answer
45 views

Problems related to conjugacy classes and their sizes.

Is $Z( G_1 \oplus G_2 \oplus G_3 \oplus\cdots\oplus G_n) = Z(G_1) \oplus Z (G_2) \oplus Z(G_3) \oplus \cdots \oplus Z$ $(G_n)$ true, where $ G_1, G_2, G_3 \cdots G_n$ are finite groups?$Z$,here refers ...
1
vote
0answers
27 views

Prove that $SL_2(F_4)$ is isomorphic to $A_5$ by giving explicit isomorphism.

Prove that $SL_2(F_4)$ is isomorphic to $A_5$ by giving explicit isomorphism. I'm really getting no idea to do this one (except I just noticed $|SL_2(F_4)|=60$ and we may relate $\mathbb{sign}$ of a ...
4
votes
3answers
107 views

Show that every group of order $35$ is abelian.

How can I show that every group of order $35$ is abelian ? I know that if such a group is abelian the it's isomorphic to $\mathbb Z_{35}$ or $\mathbb Z_7\times \mathbb Z_5$. But, how can I show that ...
0
votes
0answers
22 views

Universal enveloping group

The universal enveloping group of a monoid (with identity) is a well-known construction. If $A$ is a totally ordered set without a maximal element and $M(A)$ is the monoid of all increasing functions $...
1
vote
3answers
22 views

Misunderstanding the definition of a cycle (cyclic permutation)

Given a set $E$, let $\mathfrak{S}_E$ be the group of permutations of $E$. Definition.$\ \ $ Let $E$ be a finite set, $\zeta$ a permutation of $E$, and $\overline{\zeta}$ the subgroup of $\...
0
votes
0answers
37 views

Primitive solvable group

Let $G$ be a finite solvable group. Suppose that $G=HN$ for all minimal normal subgroups $N$ of $G$. To show that $H = G$ or $G$ is primitive If $N$ is a minimal subgroup of $G$ then $N$ is an ...
3
votes
2answers
45 views

Subgroups of finite abelian groups.

For every subgroup $H$ of a finite abelian group $G,$ there exists a subgroup $N$ of $G$ such that $G/N \cong H.$ I need to prove this or give a counter example. I am aware of isomorphism theorems ...
1
vote
3answers
31 views

Finding all normal subgroups of $A_4$

I was reading up on this: Find the number of normal subgroups of $A_4$. If $H$ has a $3$-cycle, say $(123)$, then $H$ has its inverse $(132)$ thefore it also has $(124) = (324)(132)(324)^{-1}$, ...
16
votes
2answers
617 views

Minimal generating set of Rubik's Cube group

The Rubik's Cube group is generated by the six moves $\{F,B,U,D,L,R\}$. However, is this the minimal generating set for the group? In other words, can I simulate the move $F$ just by making the moves $...
5
votes
0answers
67 views

Almost all finite groups have order $2^n$?

This might be a stupid question, but here it goes: Is anything known about, whether: $$\lim_{n\to \infty} \frac{\#\{\text{Groups of order }2^n\}}{\#\{\text{Groups of order} \leq 2^n\}} = 1$$ (where ...
0
votes
0answers
27 views

Differences among the Group Cohomology with coefficients over a commutative ring and coefficients over an arbitrary $G$-module

Assume that $G$ is a finite group and $k$ is an arbitrary commutative ring. From the general theory we know that the group cohomology $H^{*}(G ; k)$ becomes a graded ring (with the cup product). ...
1
vote
1answer
28 views

Proving that $A+B - (A \cap B) = A \cup B$ for binary integers

I hope computing questions are fine here. I'm trying to show that for all binary numbers $A$ and $B$, $A+B - (A \cap B) = A \cup B$. It's confusing me firstly because I'm not sure what the "set ...
1
vote
2answers
68 views

why are these two different in abstract algebra?

Let G be a nonempty set closed under an associative product,which in addition satisfies: (1) There exists an $e\in G$ such that $a.e=a \forall a \in G$ (2)Give $a \in G$, there exists an element $y(...
2
votes
0answers
42 views

Additivity of trace

Let $A$ be a finitely generated abelian group and $\alpha:A\to A$ be an endomorphism. Since $A=A_{free}\oplus A_{torsion}$, we can induce $\bar \alpha:A_{free}\to A_{free}$, i.e. $\bar\alpha$ is a map ...
1
vote
2answers
45 views

If $gh = hg, \ \ \gcd(|g|, |h|) = 1$, then $|gh| = |g||h|$($|a|$ is the order of element $a$ in a group $G$)

Let $G$ be a group and $g,h \in G$. I need to prove that if $g$ and $h$ commute and their orders are coprime, then $|gh| = |g||h|$, that is, the order of their product is the multiple of their orders. ...
0
votes
1answer
54 views

Some questions about a proof referring to hyperbolic group

I feel confused about: 1:It says that "otherwise H contains an infinite cyclic characteristic subgroup C," however, by definition, if H is elementary, we can only get that H contains an infinite ...
2
votes
0answers
92 views

Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?

[Update: I've now asked the same question on mathoverflow.] For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes: $$ \forall f,g,h\in G:hg(f)=h(g(f)) $$ Now suppose ...
0
votes
3answers
50 views

Ley $G$ be a group of prime order $p$. Then $|Aut(G)|=p-1$

Let $G$ be a group of order $p$ where $p$ is a prime number( hence, $G$ is cyclic ) Prove that the group of automorphisms of $G$ has order $p-1$. Since $p$ is prime, for any homomorphism $\phi: G \to ...
4
votes
1answer
60 views

Which group is isomorphic to?

If I have an abelian group $G$ of order $p^n$, how can I decide if it's isomorphic to $\Bbb{Z}_p \times \Bbb{Z}_p \times\ldots \times \Bbb{Z}_p$ ($n$ times) or to $\Bbb{Z}_{p^2} \times \Bbb{Z}_p \...
1
vote
1answer
70 views

Troubles to understanding notation and some terminology on cohomology of finite groups

I am reading Adem, Milgram's book "Cohomology of finite groups" and I have some troubles with the notation. In particular, I don't get whether they are referring to the cohomology as a ring or as a ...
0
votes
1answer
32 views

Finding elements in $S_{3}$

Question: In $S_{3}$, find elements $\alpha$ and $\beta$ such that $\left | \alpha \right |=2,\left | \beta \right |=2$ and $\left | \alpha \beta \right |=3$ I note that the permutation in $S_{3}$ ...
0
votes
0answers
8 views

Irreducible tensor for fundamental representation of SU(N=3)

I am trying to calculate the singlets of the tensor product $N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^*$. I know that $N_c \otimes N_c^*=1\oplus (...
4
votes
1answer
39 views

Does the commutator group of $S_n$ equal $A_n$ in general?

And how would one deduce this? $[S_n, S_n]$ consists of even permutations so it's obvious that $[S_n, S_n] \leq A_n$, but is $[S_n, S_n] = A_n$ true as well? If so, how to deduce this? If not, how ...
2
votes
1answer
35 views

Correlation amomg cohomology of a group $G$ and singular coohomology of its classifying space $BG$

Assume that $G$ is a finite group and let's denote by $BG$ its classifying space. Then we know that we can construct the cohomology $H^{*}(G ; M)$ with (local) coefficients, for a $G$-module $M$, in ...
0
votes
1answer
27 views

How many odd permutation of order 4 does $S_{6}$ have?

Question: How many odd permutation of order 4 does $S_{6}$ have? Possibly, there is 1 cycle of length 6-odd 1 cycle of length 4 and 1 cycle of length 2-even 2 cycle of length 3-odd 3 cycle of ...
2
votes
0answers
35 views

Orbits of the permutation action of a subgroup on its cosets

Consider a finite group $G$ and a subgroup $H \subseteq G$. There is a transitive group action of $G$ on the set of left cosets $gH$ by left multiplication, and the stabilizer of $gH$ is $gHg^{-1}$. ...
3
votes
5answers
156 views

On the definition of free products

I am a little confused about the definition of free products. Given a collection of groups $\{G_\alpha\}_\alpha$ in order to create their free product, I don't understand what properties these $G_\...
2
votes
3answers
51 views

Showing a finite abelian group is cyclic assuming something about all homomorphic images of it

Let $G$ be a finite abelian group such that $|G|\ne p^n$ for any prime $p$. If every homomorphic image $\varphi (G)$ with $|\varphi (G)| < |G|$ is cyclic, then show $G$ is cyclic. This is an old ...
5
votes
3answers
52 views

The group algebra $KG$

If $G$ is a cyclic group of order $m$. Then $KG\cong K[t]/(t^m-1)$. Where $K$ is a field. I define \begin{align*} \varphi:K[t]&\longrightarrow KG\\ \sum_ia_it^i&\longmapsto\sum_ia_ig^i \end{...
1
vote
1answer
32 views

groups as modules

Let $G$ a abelian group. Prove that $G$ is a $\mathbb{Z}$-module. Let $x\in G$ and $n\in\mathbb{Z}$ we define $xn$ as follows: If $n\geq 0$, then $x0=0$ and $x(n+1)=xn+x$. If $n<0$, then $xn=(-x)(...
4
votes
4answers
170 views

Element of infinite order for a given group presentation

Let $G=\langle a,b,c,d \mid abcda^{-1}b^{-1}c^{-1}d^{-1}\rangle$ be our presentation. The claim is that the commutator $[a,b]$ has inifinite order in $G$. I think this might be related to small ...
2
votes
2answers
52 views

what does $\ltimes$ in the context of representation theory mean?

I am considering the following sentence wich is part of a theorem: '' Let $V$ be a finite dimensional unitary representation of $H=\mathbb{Z}^{2} \rtimes $ SL$_2(\mathbb{Z})$." I have no background ...
0
votes
0answers
27 views

Coupling Homomorphism and Applications

Recently, I was reading Derek Robinson's book "A Course in the Theory of Groups" and I stumbled upon a notion which did not understand. Assume that $\thinspace$ $0 \rightarrow N \rightarrow E \...
4
votes
1answer
29 views

Showing that $x$ is an element of group $G$ by left multiplication

$G$ is a group and $H \leq G$ with $|G:H|=3$. Show that $x$ is an element of $H$ if $x \in G$ with $|x|=7$. Hint: let $\langle x \rangle$ act on $G/H$ by left multiplication and look at the orbits. ...
2
votes
0answers
82 views

Is group theory and in general pure maths useful in applied mathematics?

I've just finished the first year of Mathematics at University. I've already encountered groups, in a soft mini-course that took around half semester. My idea would be to study applied mathematics, ...
4
votes
1answer
56 views

Determine all possible $\phi$

Let $\phi: S_6 \to \mathbb{Z}/6\mathbb{Z}$ be a homomorphism. Explain why $[S_6,S_6]$, the commutator subgroup of $S_6$, is a subset of ker($\phi$) and after that determine all possible $\phi$. ...
3
votes
0answers
87 views

Integral identity graphs — smallest example

From Paulus Graphs. "The (25,2)-, (25,4)-, and (26,10)-Paulus graphs have the apparently rather unusual property of being both integral graphs (or asymmetric) and identity graphs (a graph spectrum ...
2
votes
1answer
69 views

If $G=\left<(12),(34),(45)\right>\subset S_5$, then $G\cong C_2\times S_3$

Let $G=\left<(12),(34),(45)\right>\subset S_5$. Show that $G\cong C_2\times S_3$. So my first idea was to set $a=(12)$, $b=(34)$ and $c=(45)$ and remark that $$G=\left<a,b,c\mid ab=ba,ac=ca, ...
4
votes
1answer
83 views

Number of subgroups equal to order of group

Here is a fun question. Consider the dihedral group $\mathcal{D}_4=\left\langle a,b\mid a^4=b^2=1, bab=a^{-1}\right\rangle$ of order $8$. This group has exactly $8$ genuine subgroups (but not all ...
3
votes
3answers
79 views

Can the dihedral groups be seen as subgroups of each other?

I am solving one difficult problem and now I need information on this: If $m\mid n$, is then possible to "embed" $D_m$ to $D_n$, or otherwise said, does $D_n$ have a subgroup isomorphic to $D_m$? ...
2
votes
2answers
267 views

Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
2
votes
0answers
51 views

Semi-direct product of groups

The situation is the following: Let be $G$ a locally compact (Hausdorff) group such that $G = H \rtimes_{\alpha} N$ is the semi direct product of locally compact groups $N$ and $H$. Let $A$ be a C$^*$-...
1
vote
0answers
27 views

Partial differential equation transformation

Consider the partial differential equation $$i \frac{\partial \psi}{\partial t} - \left(i\nabla + \mathbf{A} \right)^2 \psi = 0 \tag{1}$$ for the scalar function $\psi(x,y,z)$ and the vector ...
1
vote
1answer
70 views

Show that $V=\mathbb KG\oplus\cdots\oplus \mathbb KG$.

Let $V$ a $\mathbb KG$-module such that the character of $V$ is such that $\chi_V(g)=0$ for all $g\in G\setminus \{1\}$. Show that there is an $m$ s.t. $$V=\underbrace{\mathbb KG\oplus\cdots\oplus \...