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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Free Product of Groups with Presentations

There is a highly believable theorem: Let $A, B$ be disjoint sets of generators and let $F(A), F(B)$ be the corresponding free groups. Let $R_1 \subset F(A)$, $R_2 \subset F(B)$ be sets of relations ...
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32 views

Algorithmic way to check if a power-conjugate presentation is consistent?

Is there an algorithmic way to check if a power-conjugate presentation (for a finite polycyclic group) is consistent? Background: A finite solvable group $G$ has a subnormal series $$ G=G_0 ...
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77 views

Where can I find the known values for the number-of-groups-function upto $10,000\ $?

OEIS shows the number of groups of order $n$ upto $2047$. The Magma-online-calculator uses a database, but already for $1024,2004,2016,...$ it cannot determine the number of groups. Maple seems to ...
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1answer
36 views

Binary Operation in Group

The problem is actually from a high school exam (IBHL) and is supposed to be elementary: A group $G$ of order $9$ contains distinct elements $x$ and $y$ such that $x\not= y^2$. Express $(xy)^2$ as a ...
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3answers
17 views

Show that $(H, \circ)$ is a subgroup of the group $G$

Question: Let $G$ be a group and $H$ be a nonempty subset of $G$. A relation $\rho$ defined on $G$ by ``$a\rho b$ if and only if $a\circ b^{-1}\in H$" for $a,b\in G$, is an equivalence relation on ...
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3answers
623 views

Are all Lie groups Matrix Lie groups?

I have beard a bit about so-called matrix Lie groups. From what I understand (and I don't understand it well) a matrix Lie group is a closed subgroup of $GL_n(\mathbb{C})$. There is also the notion ...
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28 views

Why isn't the 2-Sylow subgroup of $D_5$ not normal when it is unique?

$D_5 = \{1, a, a^2, a^3, a^4, b, ab, a^2b, a^3b, a^4b\}$ The subgroup is $\{1, b\}$ and as far as I can tell, this is the only one. I don't understand why this isn't normal?
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Decompose $Sym^2 (V)$ into direct sum of irreducible $S_n$-subrepresentations, where $V$ is the $2$-dimensional representation

Decompose $\operatorname{Sym}^2 (V)$ into direct sum of irreducible sub representations. (Hint: Again consider the action on basis vectors.)" Here, $V=\Bbb C^2$, with its standard basis, and the ...
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1answer
41 views

Is it called the 'multiplication table' for any type of group, or only for multiplicative ones?

Suppose you had an additive group. Would the table showing its elements still be called the 'multiplication table'? If not, what is the general name given to the table showing the elements of a group ...
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58 views

Show that $\varphi (a) = a^n$ is an automorphism of $G$, if $G$ is abelian and $GCD(n, |G|)=1$

Let $G$ be abelian. Let $n \in \mathbb{Z}^+$ such that the order of $G$ and $n$ are relatively prime. Show that the function $\varphi : G \rightarrow G$ defined by $\varphi (a) = a^n $ is an ...
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If $G$ is a finite group of order $n$, then $n$ is the minimal such that $g^n=1$ for all $g \in G$?

I know that if the order of a finite group $G$ is $n$ then $g^n=1$ for all $g\in G$. But, is $n$ the smaller integer that satisfies that property? There isn't another $m<n$ such that $g^m=1$ for ...
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1answer
44 views

prove that the finite group G has a fixed point in S if |G|=$p^r$, p prime, |S|=N, (p,N)=1 [duplicate]

I am asked to solve the following Let G be a finite group with $p^r$ elements, where p is prime. if G acts on the finite set S with N elements and $(p,N)=1$, prove that there exists $s\in S$ such ...
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45 views

Finding a group of Complex numbers That is Isomorphic to this Group

So, I'm reading Introduction to Symmetry and Group Theory for Chemists to prepare for a project in Modern Algebra (and out of pure interest) and I've come across a problem that I thought would be ...
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1answer
39 views

Let $G$ be a cyclic group of order $m$, and let the number $s$ be relatively prime to $m$. Prove that if $a^s = b^s$ then it follows that $a = b$.

Let $G$ be a cyclic group of order $m$, and let the number $s$ be relatively prime to $m$. Prove that if $a^s = b^s$ then it follows that $a = b$. Let $st+mr=1$ then we have a = ...
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0answers
87 views

When solving a big Rubik cube (100x100x100), do you reduce the solution to like 50x50x50, and then 25x25x25, and then like 10x10x10 and then 3x3x3?

My question is about Rubiks cube. Say you're solving a 100x100x100 cube (you can see examples in youtube by computer program - https://www.youtube.com/watch?v=0cedyW6JdsQ) When solving a big Rubik ...
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1answer
50 views

How many elements of order $d$ are there in $\mathbf{Z}_{10}\times \mathbf{Z}_{10}$ where $d$ is a divisor of $10$?

I completed exercise $6$ page $166$ in Dummit and Foote, Abstract Algebra. It reads: "Let $G$ be a finite abelian group of type $(n_{1}, n_{2}, ..., n_{s})$. Prove $G$ contains an element of order ...
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2answers
51 views

Let $R$ be a finite integral domain, and $a$ be a nonzero element of $R$. Why is it true that there are distinct $m, n$ such that $a^m = a^n$ in $R$?

Let $R$ be a finite integral domain, and $a$ be a nonzero element of $R$. Why is it true that there are distinct $m, n$ such that $a^m = a^n$ in $R$? Does the fact that $R$ is finite imply that it ...
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1answer
62 views

Divisivility and Sylow p-subgroups

Hello I have been working in this problem for two days but i can't get the answer, I would appreciate any help or hint. Let $p$ be prime, $m\ge 1$, $r \ge 2$ and $(p,r) = 1$. If there is a ...
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33 views

Order of elements in certain generating sets of non-abelian groups !!!

The following example is just to clarify the idea. Example: The dihedral group has the following presentation $$D_{2i}=\left<s,r/s^2=r^i=e,sr=r^{-1}s \right>$$ Let $S_1=\{s,r\}$, ...
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70 views

How many elements of order 5 might be contained in a group of order 20?

Exercise from Artin's 2nd edition of Algebra. How many elements of order $5$ might be contained in a group of order $20$? My attempt: By the third Sylow Theorem, $|Syl_{5} (G)| = 1, 2, 4$ and it is ...
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40 views

Is it possible for a group to be a finite union of subgroups of infinite index?

Just restating the title: Does there exist a group $G$ and subgroups $H_1, \ldots, H_k$ so that $[G:H_i]$ is infinite for each $i = 1, 2, \ldots, k$, and $G = H_1 \cup \cdots \cup H_k$? If $G$ is a ...
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1answer
86 views

Show that the set of all groups is uncountable

Show that the set of all groups is uncountable. I think we can consider group $ ( \Bbb{R} ,+)$ and $<x>$ (generated sub group with $x)$,$x,y \in (0,1] $ now $\{<x> \cap <y> ...
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42 views

Prove H is in the center Z(G)

Exercise from Artin's 2nd edition of Algebra. Let $H$ be a normal subroup of prime order $p$ in a finite group $G$. Suppose that $p$ is the smallest prime that divides the order of $G$. Prove that ...
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1answer
54 views

Subgroups of abelian-by-finite groups

I am trying to prove that a subgroup of a abelian-by-finite group is also abelian-by-finite. I am not sure if I can use the same procedure that is used for a subgroup of a ...
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1answer
38 views

Show that a subgroup of $A_5$ that contains a 3 cycle and acts transitively on $1,2,3,4,5$ coincides with $A_5$

I know $A_5$ is simple, so my line of logic would be to prove that a group containing a 3-cycle that acts transitively on 1,2,3,4,5 must be a normal subgroup of $A_5$. I'm just not so sure how to do ...
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38 views

Which of the following group is isomorphic to the group $S_3\times \mathbb Z_2$?

The group $S_3\times \mathbb Z_2$ is isomorphic to one of the following groups: $\mathbb Z_{12}$, $\mathbb Z_6\times \mathbb Z_2$, $A_4$, $D_6$…? I know that $\mathbb Z_{12}$, $\mathbb ...
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1answer
40 views

Suppose the group satisfies below condition for b!=identity element (e) then what is b^32?

In this one I tried to first pre-multiply by a inverse and then post multiply by a , so I got the equation to be b=b ,Now how to proceed further ?
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46 views

How to prove a statement with two “ if and only if”

If $H$ and $K$ are subgroups of $G$, show that $HK$ is a subgroup if and only if $HK \subseteq KH$, if and only if $KH \subseteq HK$. This statement confuses me. Does mean I need to prove that $HK$ ...
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1answer
29 views

Let $\xi_{19} \in G_{19}$ be a primitive root of unity. Find $\Re\sum_{k=1}^9\xi_{19}^{k^2}$.

Question Let $\xi_{19} \in G_{19}$ be a primitive root of unity. Find $\Re\sum_{k=1}^9\xi_{19}^{k^2}$. Attempt I'm having doubts about how I'm solving this exercise, this is what I did: $$ ...
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1answer
29 views

Is this suffeicent to conclude it is not a subgroup

If I was wondering if $G=\{1,(123),(231),(124),(142)\}$ was a subgroup of $S_{4}$ or not could I do the following: First off notice $(123)=(231)$ so $G=\{1,(123),(231),(124),(142)\}$ And since ...
2
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1answer
31 views

Let $G$ be a group and show that $|G:Z(G)|$ cannot be prime.

Let $G$ be a group and show that $|G:Z(G)|$ cannot be prime where $Z(G)$ is the center of $G$ and $|G:Z(G)|$ is the index of $Z(G)$ in $G$. This was in a test that I had recently but I was not able ...
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Point stabliziers of primitive permutation groups are maximal primitive subgroups

While reading Kurzweil & Stellmacher's Theory of Finite Groups, chapter 6.6, Primitive subgroups are defined: a group $M$ is primitive if for every If for every non-trivial $N \unlhd M ...
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1answer
29 views

Determining if a semigroup or monoid can extend to a group

I was thinking a few days ago, and it crossed my mind to wonder if one could extend, say, an arbitrary monoid to a group. My first thought was that it ought to be simple: Take a monoid $M$ with ...
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Progressed A problem on semidirect products where one component is cyclic : a specific problem I managed partially but stuck on the rest

I was recently presented this in my abstract algebra class and I have managed some of it on my own the rest is still a mystery: Let $ H $ be a group and $ K = \langle x\rangle $ be a cyclic group ...
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1answer
44 views

Divisible group $G\neq 0$ is not free

How do I show that a divisible group $G\neq 0$ is not free? I know that divisible means that for all elements $g$ in an abelian group $G$ and $n\in\mathbb{N}$ there is an element $a\in G$ such that ...
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28 views

How to find the complement $\mathbb{Z}^3 \setminus H$

For an old exam question in Group Theory, there was the following question: Let $H \subset \mathbb{Z}$ be the group $\mathbb{Z}\begin{pmatrix} 1 \\ 2 \\ 3\end{pmatrix} + \mathbb{Z}\begin{pmatrix} 2 ...
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Uniqueness of the inverse in a group

Does it follow from the group axioms that each element has exactly one inverse? I.e. if $x$ is given then there is only one $x^{-1}$ for which $xx^{-1}=e$.
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Is this proof of $(xy)^{-1}=y^{-1}x^{-1}$ valid?

Proof. Let $G$ be a group, and $x,y\in G$. Then, we have $xy(xy)^{-1}=e$, where $e$ is the identity element of $G$. Multiplying on the left by $x^{-1}$, we get $y(xy)^{-1}=x^{-1}$. Multiplying once ...
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1answer
34 views

Prove the following set endowed with the binary operation is an abelian group

Let $∗$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x ∗ z = y$. Show that this set with the operation ...
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1answer
119 views

Let $R$ be a commutative ring with unity $1$. An element $a \in R$ is called to be nilpotent if $a^n = 0$ for some positive integer $n$

Let $R$ be a commutative ring with unity $1$. An element $a \in R$ is called nilpotent if $a^n = 0$ for some positive integer $n$. Prove that if $a, b$ are nilpotent, then so is $a + b$. Prove that ...
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The groups with nilpotent hall $p'$ subgroup.

Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elemets. Theorem $2$: A group of order $p^nq^m$ is solvable. Theorem $1$ depends on ...
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23 views

Proper mathematical description for outer perfect shuffling

I was given the following problem: Consider a pack of $2 n$ cards, numbered from 0 to $2 n − 1$. An outer perfect shuffle is a shuffle of the cards, in which one first splits the pack in two ...
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Describe all the possible images of $f$ such that $f\colon D_{10}\to H$ is a homomorphism

Let $H$ be a group and suppose that $f\colon D_{10} \to H$ is a homomorphism. Describe all the possible images of $f$. I know by definition, $\operatorname{Im} (f) = \{h \in H \mid h=f(g) \text{ for ...
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Structure theorems for infinitely generated Abelian groups

The classification theorem for finitely generated Abelian groups is well known and plays big role in mathematics. Are there any structure theorems about infinitely generated Abelian groups known?
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201 views

Is there a standard name for this infinite group?

Consider the group of sequences $$\{(a_1,a_2,\dots): a_i\in\mathbb{Z}/2\mathbb{Z}\}$$ where the group operation is component-wise addition. Is there a standard name for this group, such as ...
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5answers
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What does it mean when two Groups are isomorphic?

I'm not asking for the formal definition I know it. An isomorphism is a bijective homomorphism. In my book it's indicated many times when two groups are isomorphic, and I don't understand what's the ...
0
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1answer
83 views

Find number of ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{28}$ [duplicate]

Find number of ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{28}$. This question has been asked before but I found the solution confusing.Please check whether my approach is valid. A ring ...
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2answers
49 views

Let $G$ be the group $S_4\times S_3$ .Prove that $G$ has a normal subgroup of order $72$.

Let $G$ be the group $S_4\times S_3$ .Prove that $G$ has a normal subgroup of order $72$. Attempt:Order of $G$ is $144=2^4.3^2$.So $G$ has a Sylow $2$ subgroup of order $16$ and a Sylow $3$ subgroup ...
2
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1answer
30 views

Group Theory and closure of a group

I am trying to find subgroups of $D_{10}$. I can easily find the subgroups of order $2$ - $\{e, t\}$, $\{e, st\}, \ldots$ etc. where $s$ is the rotation of each vertices and $t$ is the reflection ...
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1answer
64 views

Is there a number $n$, such that there are $22$ groups of order $n$?

Denot : $N(n)$ : the number of groupfs of order $n$ ? Is there a number $n$ with $N(n)=22$ ? Checking the first about $2000$ numbers, I noticed that there is no $n\in [1,2000]$ with $N(n)=22$. ...