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.

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9
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1answer
63 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 $T_n v = 0$. ...
3
votes
1answer
468 views

Groups of order $pq$ have a proper normal subgroup

I am doing the following exercise from [Birkhoff and MacLane, A survey of modern algebra]: Let $G$ be a group of order $pq$ ($p,q$ primes). Show that either $G$ is cyclic or contains an element ...
2
votes
2answers
12 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 $S_n$ ...
2
votes
0answers
81 views

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

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 there is additional axiom, or constraint if you prefer, ...
15
votes
4answers
749 views

Maximum number of Sylow subgroups

I've been studying Sylow-$p$ subgroups, and I've come across this problem. Let $G$ be a finite group. Show that the number of Sylow subgroups of $G$ is at most $\frac{2}{3}|G|$ . ($|G|$ is the ...
18
votes
7answers
6k views

How to show every subgroup of a cyclic group is cyclic?

I'm teaching a group theory course now, and I wanted to give my students a proof that every subgroup of a cyclic group is cyclic. The easiest way I could think to do this is to say that any cyclic ...
1
vote
0answers
28 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 ...
0
votes
2answers
568 views

Conjugacy classes in $ D_4$

Let G be group of all symmetries of square. Find number of conjugate classes in G. I tried this question just as we do for $S_n$ that the number of conjugate classes in $S_n $ is partition number of ...
2
votes
1answer
30 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
26 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
17 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
0answers
32 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 $gHg^{-1}$. ...
1
vote
3answers
665 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
2answers
28 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
43 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 {...
0
votes
2answers
21 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 ...
3
votes
3answers
66 views

An erroneous application of the Counting Theorem to a regular hexagon?

I'm trying to count the unique orbits of a regular hexagon such that each vertex is either Black or White and each edge is either Red, Gree, or Blue. The group I've chosen to act on the hexagon is the ...
-5
votes
1answer
216 views

Let $G$ be finite group. if $A,B\le G$ with orders $4, 5$ respectively then $A \cap B$? [closed]

Let $G$ be finite group. If $A$ and $B$ are subgroups of $G$ with orders 4 and 5 respectively, what is $A \cap B$ ?
2
votes
1answer
30 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 ) ...
0
votes
1answer
36 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 ...
16
votes
2answers
587 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 $...
0
votes
1answer
33 views

Order of product of non-disjoint cycles

Let $a$ and $b$ be two non-disjoint cycles of order $m$ and $n$. Is there any general formula for the order of $a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ...
0
votes
1answer
40 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 | ...
-1
votes
0answers
32 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
0answers
23 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
102 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 ...
3
votes
1answer
38 views

Permutations of $S_7$

Find all permutations $\alpha \in S_7$ such that $\alpha^3 = (1 2 3 4)$. My attempt: We know that such an $\alpha$ must "look like" $(1432)$, since $(1432)^3=(1234)$. I think I need to find the ...
2
votes
2answers
214 views

Normal subgroup in center of the group

Let $G$ be a group of order $3825$. Prove that if $H$ is a normal subgroup of order $17$ in $G$ then $H\leq Z(G)$. In the link below, the solution basically says that the index of $C_{G}(H)$ must ...
0
votes
0answers
34 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 ...
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
20 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 $...
4
votes
4answers
167 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 ...
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 $(124) = (324)(132)(324)^{-1}$, ...
29
votes
2answers
2k views

Rubik's Cube Not a Group?

I read online that although the 3x3x3 is a great example of a mathematical group, larger cubes aren't groups at all. How can that be true? There is obviously an identity and it is closed, so ...
3
votes
2answers
43 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 ...
5
votes
0answers
51 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
24 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). ...
2
votes
3answers
50 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 ...
2
votes
2answers
74 views

Sylow subgroup of a product of subgroups

Let $UV$ be a subgroup of $G$ and $P$ is a Sylow $p$-subgroup of $G$. Suppose that $P \cap U \in \operatorname{Syl}_p(U)$ and $P \cap V \in \operatorname{Syl}_p(V)$. I need to show that $P\cap UV \in \...
0
votes
1answer
50 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
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
1answer
26 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(...
1
vote
2answers
60 views

Aut$(K/F)$ permutes roots of polynomial.

Let Aut$(K/F)$ is the set of all automorphism from $F$ to $K$, where $K$ is a galois extension of $F$. Let $f(x) \in F[x]$ and $\alpha$ be a root of the polynomial $f(x)$. I am able to prove that for ...
1
vote
1answer
621 views

General linear group/special linear group is isomorphic to R*

Let $GL(n,\mathbb{R})$ be the group of invertible $n \times n$ real matrices, let $SL(n,\mathbb{R})$ be the group of $n \times n$ real matrices of determinant $1$, and $\mathbb{R}^*$ be the group of ...
1
vote
2answers
43 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
3answers
48 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 ...
3
votes
1answer
56 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
40 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 ...