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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106 views

Every group of order $150$ has a normal subgroup of order $25$

Let $G$ be a group of order $150$. I must show that it has a normal subgroup of order $25$. The hint says to show that is has a normal subgroup of order $5$ or $25$. Now from Sylow, I know that the ...
6
votes
1answer
122 views

Extension of group with Ext$^{1} (A, B) = 0.$

Are there any infinite torsion free abelian groups $A$ and $B,$ with $A$ is not projective and $B$ is not divisible but $$\text{Ext} ^{1}(A, B) = 0.$$ Thanks
1
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1answer
28 views

non-cyclic example that there is an element not contained in any maximal subgroup

I've just learned today that the analogy of the theorem "every non-unit element in a ring with identity is contained in a maximal ideal" is not true for groups, that is, there are some groups that ...
4
votes
1answer
68 views

Why is it true that $|AB:A|=|B:A\cap B|$ even if $A$ is not normal in $AB$? (Second Isomorphism Theorem)

I just read about the First and Second Isomorphism Theorems in the book Abstract Algebra by Dummit and Foote. After proving the Second Isomorphism Theorem, they said: Proposition 13 isn't ...
1
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3answers
66 views

Well-defined map from $G/N$ to $G$ that is a homomorphsim?

Let $G$ be a group, and let $N$ be a normal subgroup of $G$. I know that the natural projection homomorphism is a surjective homomorphism from $G$ onto $G/N$. If I choose a particular representative, ...
2
votes
2answers
53 views

Normality of a normal subgroup of normal subgroup of G

Let $G$ be a non-Abelian Group and $H$ is normal subgroup of $G$. Is it always true that a normal subgroup $K$ of $H$ is also normal in $G$? Justify your answer. My answer is that, this is not true ...
3
votes
3answers
33 views

why do we need to demand the closure condition on the definition of a group?

so far, this is the definition I saw in number of sources: A group $(G,·)$ is a nonempty set $G$ together with a binary operation $\cdot$ on $G$ such that the following conditions hold: (i) ...
0
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1answer
34 views

Why is the orbit $\cal{O}_x$ of the circle group a circle with radius $|x|$?

Let $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$. What is the orbit $\cal{O}_x$ of $x\in \mathbb{C}$ under $\mathbb{T}$?
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0answers
38 views

General form of a matrix $M$ commutes with the unitary representation $U^{\otimes m},~ \forall U\in U(n)$

My question is about the general form of a $n^m\times n^m$ positive definite matrix $M$ where $$[M,U^{\otimes m}]=0,~ \forall U\in U(n)$$ or in other words, M commutes with all members of the the ...
-1
votes
2answers
35 views

Prove a cycle of length l is odd if l is even? [closed]

This is my first course on Group Theory. How do I go about proving this?
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0answers
19 views

Find a particular order of elements in group of units modulo $100$. [duplicate]

Let $\mathbb Z^*_{100}$ be a group of units modulo $100$, i.e., $\mathbb Z^*_{100}=\{1,3,7,9,13,17,19,\dots,97,99\}$. Find such $a\in \mathbb Z^*_{100}$ such that the order of $a$ is $20$ in $\mathbb ...
4
votes
2answers
49 views

Showing Sylow $p$-groups are Abelian if $n_p=2p+1$

An old qual question on Sylow $p$-groups: Assume that $p$ is an odd prime and $G$ is a finite simple group with exactly $2p+1$ Sylow $p$-groups. Prove that the Sylow $p$-groups of $G$ are abelian. ...
2
votes
1answer
67 views

Prove G is Abelian

Let $G$ be a group and $a,b\in G$. given that $(ab)^k=a^k b^k$ and $(ab)^{k+2}=a^{k+2} b^{k+2}$ for some $k\in \mathbb N$. prove that $G$ is abelian. So far my attempt was: ...
0
votes
0answers
21 views

What is the purpose of the almost maximal and $ p $-supersoluble subgroup?

Suppose that $ H \unlhd G $ such that $ G/H $ is supersoluble. Suppose that there is an element $ y \in H $ such that $ H = \langle y \rangle L $ for any almost maximal subgroup L of $ H $; then $ G $ ...
0
votes
1answer
29 views

Sylow Theorems Problem..

Suppose $G$ is a simple group of order $315=3^2\times 5\times 7.$ Then I can prove that $G$ has $84$ elements of order $5$ and $90$ elements of order $7.$ But when we consider the Sylow $3$ ...
6
votes
4answers
59 views

how do I prove that $S_4$ has no normal subgroup of order 6

Let $N$ be a normal subgroup of $S_4$. I have proven that $|N|\ne 2,3,8$. Yet, I don't know how to prove that $|N|\neq 6$. Should I compute all subgroups and check this?
3
votes
1answer
31 views

Presentation of groups and positive expressions

For a group $G=〈S|R〉$, $S,R$ are both finite. A positive element $g\in G$ is an element of $G$ that can be written as a finite product of elements of $S$ only. A positive expression of $g$ is a word ...
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votes
1answer
46 views

Group $G$ commutes only with the identity element and itself [closed]

Let $S$ be the collection of (isomorphism classes of) groups $G$ which have the property that every element of $G$ commutes only with the identity element and itself.Then which one is correct? A. ...
5
votes
1answer
46 views

How do we identify $\mathfrak{R}$-automorphisms of a group?

If $G$ is a finite group, a bijection $f\colon G\to G$ is called a (normed) $\boldsymbol{\mathfrak{R}}$-automorphism if $f$ maps subgroups of $G$ to subgroups of $G$, and $f(gH) = f(g) f(H)$ for any ...
2
votes
0answers
34 views

Different definitions of projective matrix groups, with one giving an algebraic group but not the other

Recently my professor told me that the usual definition of PSL(2,$\mathbb{R}$) = SL(2,$\mathbb{R}$)/{$\pm$I} does not give an algebraic group, but the following definition does: PSL(2,$\mathbb{R}$) ...
6
votes
1answer
41 views

Median order of an element in an additive group modulo $n$

I'm trying to gain some intuition here. Suppose we have the group $\mathbb{Z}_{n}$ (with the operation being addition modulo $n$). What is the median order of an element of this group when $n$ is a ...
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1answer
34 views

Short exact sequences and different extension

Let $A = \mathbb Z$ and $B = \mathbb Q.$ Then Ext$(A, B)$ gives the set of all equivalent extensions of $A$ by $B.$ I have few questions. Is this sequence $0\rightarrow \mathbb Z ...
2
votes
1answer
50 views

Matrix and Abelian groups question

Let $A$ be a Matrix: $$ A=\begin{pmatrix} 1 & 2\\ 4 & 1 \end{pmatrix} $$ Let $f\colon v\to Av$ be a homomorphism from $Z^2$ to $Z^2$. Find a base $(v_1,v_2)$ to $Z^2$ and $2$ integers ...
2
votes
1answer
59 views

Find total number of elements of order 20 in the multiplicative group $\mathbb Z^*_{100}$

How can I find all the elements of order $20$ in the multiplicative group $\mathbb Z^*_{100}$. $[7]\in \mathbb Z^*_{100}$. $7^4\equiv 1\pmod{100}$. So order of $[7]=4$. But how can I find all ...
2
votes
3answers
50 views

find all odd permutations $\sigma \in S_4$ such that $\sigma (123) \sigma^{-1} = (234) $

need help with this question... find all odd permutations $\sigma \in S_4$ such that $\sigma (123) \sigma^{-1} = (234) $ really have no idea how to approach this. thanks.
0
votes
1answer
32 views

Question from book 'Indra's Pearls' about limit set arising from infinite words (compositions of maps)

The book considers mappings $a, b, A,$ and $B$ where $A = a^{-1}, B = b^{-1}$. It goes on to say that words represented by compositions of these maps (e.g. $abbA$) correspond to points. I ...
0
votes
0answers
12 views

Constructing Følner sequence from invariant mean with prescribed density on a given set.

Let $G$ be a discrete countable amenable group. Suppose that $\lambda$ is an invariant mean on $G$ and $B \subset G$. Does there exists a Følner sequence $F_n \subset G$ such that $$\lambda(\chi_B) = ...
1
vote
2answers
69 views

Set of generators for $A_n$, the alternating group.

The problem is this: Prove that $A_n = \langle (123),(124),\ldots,(12n)\rangle$. I had cogitated this problem for quite awhile, and haven't been able to come up with anything. The only good idea ...
2
votes
3answers
51 views

Which inverse multiplicative groups modulo $n$ are cyclic or not

I've found nothing about this in my book neither in the internet. Also the wikipedia article about inverse multiplicative modulo $n$ is poor. So, I need prove that $$\mathbb Z_n^*$$ is cyclic for ...
1
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0answers
46 views

Can anyone explain these conclusions? Permutations, Symetric group…

The conclusions start off like this:I will highlight what is unclear in yellow. $sgn G$-sign of G permutation, $Ker$-kernel of a function Lets define the function: $\ \Phi$ like: $(\forall G \in ...
0
votes
2answers
50 views

Exercise about finding group isomorphisms

So, i've just learned about groups homomorphisms and isomorphisms. I know that an homormorphism betweet two groups is a function such that $$\phi(a\square b) = \phi(a)\star \phi(b)$$ And when the ...
1
vote
2answers
53 views

Subgroup of a group with 24 elements.

Suppose that $G$ is a finite group of order $24$, which has four $3$-sylow subgroups. We know that may contain $1$ or $3$ 2-sylow subgroup. How can I prove that there only exists one $2$-sylow ...
3
votes
2answers
124 views

Consequence of First Homomorphism Theorem?

Let $\phi:G\to\bar G$ be a surjective homomorphism with kernel $N$. Then the first homomorphism theorem tells us that $G/N\cong\bar G$. My question is this: Lagrange's theorem also tells us that ...
3
votes
1answer
107 views

A question on subgroups of a finite group

Let $G$ be a finite group. Let $K$ and $H$ be subgroups of $G$, where $K$ is normal in $G$ and $\gcd([G:H],|K|)=1$. Prove that $K$ is a subgroup of $H$. So far we found that $o(K)$ divides ...
3
votes
2answers
38 views

Dihedral subgroups of $S_4$

Prove that in $S_4$ there are $3$ groups that are isomorphic to $D_4$. I know that the $2$-sylows of $S_4$ should be subgroups of order $8$, but to prove it is a bit tricky for me Any help ...
1
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1answer
20 views

Counting symmetries using elementary method

I am studying group theory using Armstrong's Groups and Symmetry, one of the biggest problem is that there is no solution manual available. Thus I will rely on you guys! Find all the rotational ...
4
votes
4answers
61 views

$G$, group. $a,b\in G$. Show that $|a|=|a^{-1}|; |ab|=|ba|$, and $|a|=|cac^{-1}|, \forall c\in G$. [duplicate]

Isn't it obvious for $|a|=|a^{-1}|$, since $\langle a\rangle = \langle a^{-1} \rangle$? For $|ab|=|ba|$, I think we should go like this: $e=(ab)^n\Rightarrow e=(ab)(ab)^{n-1}\Rightarrow ...
2
votes
1answer
35 views

Let G be a group, $N<K<G$ and $N\trianglelefteq G$. Prove that $K/N \trianglelefteq G/N$

Let G be a group, $N<K<G$ and $N\trianglelefteq G$. Prove that $K/N \trianglelefteq G/N$ What I have tried is: Note that $1\in G$. So $1\in K$ and $N1=N\in K/N$ which shows that $K/N$ is ...
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2answers
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Finite Abelian Groups question [closed]

Let A be a finite abelian group. Let m be the smallest natural number such that ma=0 for every a in A. Prove that there is an element in A of the order m
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2answers
33 views

Show that left cosets partition the group

I know how to prove that it happens, by proving that the left coset definition actually is an equivalence relation. Then, it's proved that it partitions the set, since equivalence relations do it. ...
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0answers
20 views

A graph and metric on the class of finitely presented groups

Pick some countably infinite set $Y$. Call two finitely presented groups $G_1$ and $G_2$ presentation related if for some finite $X \subseteq Y$, $G_1 \cong F\{X\} / H$ and $G_2 \cong F\{X\} / ...
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votes
2answers
82 views

Prove that $H$ is a normal subgroup of $G$

This is a problem from the book "Berkeley Problems in Mathematics": Let $G$ be a group of order $120$, let $H$ be a subgroup of order $24$, and assume that there is at least one coset of $H$ ...
2
votes
1answer
33 views

The non-functional number

Lets say $f(1)=1, f(2)=2, f(3)=3,$ and $f(4)=n$. Some rules to follow are that: $1)$ $f(x)$ could literally be any function and it depends on what $n$ is. 2) $n$ is an integer. 3) If there is an ...
0
votes
1answer
48 views

If $G$ is a group and $a,b \in G $, then: $ |\langle a,b \rangle|=|\langle a\rangle|\cdot |\langle b\rangle| \ \Longleftrightarrow a=b$?

Such that $\langle a\rangle , \langle a,b\rangle $ is a subgroups of $G$ generated by $\lbrace a\rbrace, \lbrace a,b\rbrace $ respectively and $| \ . |$ is the member of $\langle a\rangle $ ...
4
votes
1answer
76 views

A group satisfying $G=[G,G]$?

What kind of non-identity group $G$ satisfies $G=G'=[G,G]$? How is it related to a solvable or nilpotent group? Thanks in advance.
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0answers
28 views

If $ G $ is a finite group. Suppose $ 1 = G_{0} \unlhd G_{1} \unlhd \cdots \unlhd G_{s} = G $ is a chief series of $ G $.

The $ p $-Fitting Subgroup of $ G $ is the maximal normal $ p $-nilpotent subgroup Of $ G $ and write it $ F_{p}(G) $. If $ G $ is a finite group. Suppose $ 1 = G_{0} \unlhd G_{1} \unlhd \cdots ...
3
votes
0answers
29 views

Which abelian groups have only a single composition series?

Cyclic groups of composite powers don't: for example, $1=C_1\triangleleft C_3\triangleleft C_6 $ and $1=C_1\triangleleft C_2\triangleleft C_6 $ are both composition series for $C_6$. But cyclic ...
2
votes
2answers
34 views

A question about Coxeter groups.

Let $G$ be a group, and $x,y,z \in G - \{1\}$, where $1$ is the identity element of $G$. Assume that $x^2=y^2=z^2=1$, $xy =yz$ and $xz$ is of infinite order. Can $G$ be a Coxeter group? Can $G$ be ...
1
vote
1answer
19 views

How to generate the icosahedral groups $I$ and $I_h$?

The icosahedral groups $I$ with 60 elements and $I_h = I \times Z_2$ are also three dimensional point groups. However, ever unlike other point groups, it seems there is rarely reference to give their ...
0
votes
1answer
19 views

An inductive limit of amenable groups is amenable

It is a Theorem that an inductive Limit of amenable Groups is amenable. Could someone sketch me a proof of this, or give me a reference? I couldn't find one. Thanks in advance. Edit: I wanted it for ...