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)

0
votes
1answer
18 views

Show how (0,(12)) and (1,(12)) are in different conjugacy classes.

$S_3=${${(1),(12),(13),(23),(123),(132)}$}. $\mathbb Z_2=${${0,1}$}. $\mathbb Z_2$$\oplus$$S_3$={$(0,(1)),(0,(12)),(0,(13)),(0,(23)),(0,(123)),(0,(132)),(1,(1)),(1,(12)),(1,(13)),(1,(23)),(1,(123)),(...
1
vote
2answers
36 views

$\mathbb{Z}$ has no composition series. Need an assistance in some questions.

Please, read the whole post before trying to answer. Remark: here $\subset$ means "strict inclusion". I need to prove that the group $\mathbb{Z}$ has no composition series. That is, no normal series ...
0
votes
0answers
16 views

Let $G=AB$ where $(|A|,|B|)=1$ and $V$ be an $\mathbb{F}[G]$ module.

Under these assumptions it is a well-known fact that if $V_A$ and $V_B$ are faithful ($V_A$ denotes $V$ as an $\mathbb{F}[A]$-module) then $V$ is also faithful. Clearly if $V_A$ and $V_B$ is ...
-1
votes
0answers
40 views

Class-equation of $\mathbb Z_2$ $\oplus$ $S_3$.

$S_3=${${(1),(12),(13),(23),(123),(132)}$}. $\mathbb Z_2=${${0,1}$}. $\mathbb Z_2$$\oplus$$S_3$={$(0,(1)),(0,(12)),(0,(13)),(0,(23)),(0,(123)),(0,(132)),(1,(1)),(1,(12)),(1,(13)),(1,(23)),(1,(123)),(...
1
vote
0answers
15 views

Intersection of the kernel of the irreducible characters determinants

Let $G$ be a finite group. It is easy to show that $G'\le \bigcap_{\chi\in Irr(G)}Kerdet\chi$. Is there equality ? This question arises from the remarkable equalities $\bigcap_{\chi\in Irr(G)}Ker\...
1
vote
0answers
41 views

Determinant of a character

let two characters $\chi$ and $\vartheta$ of a finite group $G$ (assumed to be non-null). Let $\mathfrak{X}$ and $\mathfrak{Y}$ be representations of $G$ affording respectively $\chi$ and $\vartheta$ ...
1
vote
2answers
37 views

Are the groups $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices and $O(n,\mathbb R)$ isomorphic?

Let $G:=\{A \in M(n,\mathbb R) : A=A^t\}$ i.e. the group ( under addition ) of symmetric matrices ; Are $G$ and $O(n,\mathbb R)$ isomorphic ?
2
votes
2answers
26 views

A simple group such that $[G:H]=n$ can be embedded into $A_n$

Let $G$ be a finite simple group and $H$ be a proper subgroup of $G$ such that $|G:H|=n$. Then, how do I prove that $G$ can be embeded into $A_n$? I can prove that $G$ can be embedded into $S_n$ ...
1
vote
0answers
39 views

Recent advancement in Haar measure

From my personal interest I have studied Haar Measure and the related concept of group theory on my own. However due to the lack of an authoritative source it is not getting possible for me to know ...
1
vote
2answers
28 views

Why if $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic? [duplicate]

In the notes I'm studying from ( again =) ) I read: If $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic Could someone give me a justification for ...
2
votes
2answers
18 views

Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
2
votes
1answer
41 views

On the maximum number of Sylow subgroups

I have some doubts regarding the first sentences below the fourth case in the more elementary solution by Prof. Samuel to problem 11856 of the Monthly. Here you have a screenshot of the relevant part ...
0
votes
2answers
30 views

$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?

Let $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then is it true that $G=HK$ ? ( I know that the fact is true if $p=2$ ...
1
vote
1answer
52 views

A proposition about free product

My question is about a proposition that I found on Munkres book: Topology (page 419). My definitions are based on this book. I give you the definition of free product that he uses. $\textbf {...
2
votes
3answers
28 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
81 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
685 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 | ...
2
votes
1answer
33 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
33 views

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

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
44 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
26 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
106 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
36 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
44 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
30 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
607 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
59 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
26 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
67 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
53 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
91 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
49 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
67 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
37 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
33 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
154 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_\...