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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$|G|=p^2$ then $G \cong \mathbb Z_{p^2}$ or $G \cong \mathbb Z_{p} \times \mathbb Z_{p}$

Problem Let $G$ be a group with $|G|=p^2$ for some prime $p$, then $G \cong \mathbb Z_{p^2}$ or $G \cong \mathbb Z_{p} \times \mathbb Z_{p}$. I think I came up with a solution to this problem but I ...
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70 views

Showing a Group $G$ is not Simple [duplicate]

Let $G$ be a finite group of order $pq$, where $p,q$ are distinct prime numbers. Show that $G$ is not simple. Here is my attempt: $|G|=pq$. If $G$ is not simple, then it has non-trivial subgroups, ...
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50 views

Clarification on Symmetry Group

My text says "It is a general fact, and an easy one to prove, that the invertible transformations of a mathematical object that preserve some feature of its structure always form a group. We call ...
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1answer
76 views

A question in Abstract Algebra about cosets

I tried to solve this problem but without success: Let $H$ be a subgroup of a group $G$. Build an injective function from $G/H$ (the set of left cosets of $H$) to $H\setminus G$ (the set of right ...
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1answer
41 views

A specific group lacking Følner sequences

How does one go about proving that the free group $<a,b,a^{-1},b^{-1}>$ lacks any Følner sequence?
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47 views

How to show $\operatorname{GL}_2 (\mathbb Z_2)\cong D_3$

I want to show that $\operatorname{GL}_2 (\mathbb Z_2)\cong D_3$ while $\operatorname{GL}_2(\mathbb Z_2)$ is the group of matrices $2\times 2$ above $\mathbb Z_2$. I tried to show that maybe every ...
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2answers
55 views

Why are the two definitions of a generating set of a group equivalent?

I know this is a simple question, but I'm really bad at math. Definition 1: $\left<S\right>$ is a subset such that every element of the group can be expressed as the combination (under the ...
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2answers
146 views

Prove that $G$ has a subgroup isomorphic to $G/H$.

Let $G$ be a finite abelian group of order $n$ and let $H$ be a subgroup of $G$ of order $m$. Show that $G$ has a subgroup isomorphic to $G/H$. Here are my thoughts: Define $\mu_n := \{z \in ...
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50 views

Subgroup of a p-group in its center

Suppose $p$ is prime, $n\in\mathbb Z^+$, and $G$ is a group of order $p^n$. If $H$ is a subgroup of order $p$ and $ghg^{-1}\in H$ for all $g\in G$ and $h\in H$,I can't seem to show that $H \subseteq ...
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171 views

Prove that the unity element in a subfield of a field must be the unity of the whole field

The solution I was given says: Let $F$ be a field and suppose $u^2=u$ for some nonzero $u$ in $F$. By multiplying each side by $u^{-1}$ it is clear that $0$ and $1$ are the only solutions of ...
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Prove that stabilizer subgroups of G are conjugate to each other

Suppose that a group $G$ acts on a set $X$. Show that if $x_1$ and $x_2$ in X are in the same $G$-orbit, then their stabilizer subgroups of $G$ are conjugate to each other. My proof: Assume ...
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156 views

Group Action problem - Proving an action and finding number of orbits

The following question is from Serge Lang's Undergraduate Algebra. I have trouble in finishing part a and understanding part b. Let $S, T$ be sets and let $M(S,T)$ denote the set of all mappings ...
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1answer
109 views

Is there a canonical finite group multiplication table?

If one were to lay out the dihedral group $D_4$ multiplication table, it might look like this (generators $a$ and $b$):       If we abstract from the generators and just call the ...
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48 views

Group theory problem automorfism

Let $G$ be a finite group. If there exist an automorphism $f$ such that if $f(x)=x \iff x=e$ and $f(f(x))=x$ for all $x$ in $G$, then prove $G$ is Abelian.
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Classifying $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} / <(3,3,3)>$

Again, my instructor's explanation in class is so confusing to me. Here we cannot take the original approach of finding the order and limiting our choices. So instead my instructor says that this is ...
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71 views

Show there is a homomorphism from $G/N$ onto $K$.

Let $\sigma: G \to K$ be an epimorphism (onto homomorphism). And let $N$ be a subgroup of $\ker( )$ that $N \triangleright G$. Show there is a homomorphism from $G/N$ onto $K$. Note that if ...
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66 views

How to check which subgroups of $D_4$ are normal

How do I check which subgroups of $D_4$ are normal? Trying all elements seems very cumbersome. So far, I know only basic theorems like Lagrange's and the homomorphism as well as the isomorphism ...
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2answers
45 views

Compatibility of homomorphisms and quotient maps of abelian groups

Suppose $A$ and $C$ are abelian groups with subgroups $A'$ and $C'$ respectively. Let $f:A\to C$ be a group homomorphism. I was wondering if the following statements are equivalent: There exists a ...
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3answers
100 views

A question about the proof of a theorem in Representation theory of groups

My Question is about one part of the proof of theorem in the book "A Course in the Theory of Groups" by Derek J.S. Robinson. I highlight the part that my question is about. We know that if $G$ is a ...
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2answers
83 views

What is this group explicitly?

Let $G$ be finite group act on a set $X$ transitively. I already proved the set $\{ f : X \to X | f(g*x) = g*f(x) \, \forall x \in X, g \in G \}$ is a group. My question is what is this group ...
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42 views

Proof for a subgroup $H$ of the finite subgroup $G$ there are at least $|H|$ elements that are not in conjugates of $H$

A problem in Isaac's finite group theory is to prove if $G$ is a finite group and $H$ is a proper subgroup of $G$ then there are at least $|H|$ elements that are not of the form $ghg^{-1}$ with $g\in ...
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66 views

Show that $D_n$ is a subgroup of Perm($\mathbb{C}$).

For $ n \in \mathbb{N}$ and $0 \le r <n$, define $f_r : \mathbb{C} \to \mathbb{C}$ ; $z \mapsto ze^{2\pi i r/n}$ and $c: \mathbb{C} \to \mathbb{C}$ ; $z \mapsto \bar{z}$. a) Let $D_n = \{ f_0, ...
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80 views

Prove $f(x) = x * a$ is bijective (preferably using inverse)

I have this question that I am stuck at. Let $(G; * ; I)$ be a group and let 'a' in $G$. Let $f : G \rightarrow G$ be the function defined by $f(x) = x * a$ for all $x$ in $G$. Prove that $f$ is ...
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108 views

How to see that $PSU(2)$ is same as $SO(3)$?

Some background: We have an action of $SU(2)$ on the space of traceless Hermitian matrices, $\mathcal{H}$, via conjugation: $$SU(2)\times \mathcal{H}\to \mathcal{H}, \ (U,H)\mapsto UHU^{-1}.$$ The ...
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75 views

Finite Group and normal Subgroup

Let $d,m\in \mathbb{Z}$ with $d,m\geq 1$ and $\gcd(d,m)=1$. Let $G$ be a group of order $dm$ and define the set $X:= \{g\in G | g^d=1\}$. Show: if $H$ is a normal subgroup of $G$ with order d then ...
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274 views

Cyclic group generators

My question is: Can you find a cyclic group with n generators? I know that zero (or any other identity element for that matter) is included, so there would be for $Z_n$ at most n-1 generators. ...
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1answer
83 views

Rotations in higher dimensions vs. Spheres

Where my questions stem from: When we study the rotations in a plane or of some specific higher dimensions, there exists a neat approach to represent all the rotations as a spheres $\mathbb S^i$, for ...
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finding all the left cosets

This is a homework problem: Determine if the index $[G:H]$ is finite or infinite. List all the left $H-$ cosets in the following: $a.$ Let $G=\mathbb{Z}\times \mathbb{Z}, \ H=\{(x,x):x\in ...
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Concerning Cyclic Groups

I am new to group theory. I have a problem but I don't really understand what it is about, so I am asking somebody to explain what is the problem (I am not really seeking for solution). Here it is: ...
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2answers
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Find the order of the elements in the given groups

I have to find the order of the following elements in the given groups: $(1 \ \ 2 \ \ 3) \ (1 \ \ 2\ \ 4) \text{ in } S_5$ $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 3 & 4 & 1 ...
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123 views

the number of all orders of elements in HS (Higman-Sims group) with GAP

I want to calculate the number of all orders of elements in HS (Higman-Sims sporadic simple group). Is there any way of doing this with MAGMA or GAP? How I can determine orders of elements of a group ...
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112 views

Show that there are infinitely many primes $p$ such that $p = 1 (\mod q)$ in a very specific way

I friend of mine has shown the following: Let $n \in \mathbb{N}$, and $q$ an odd prime number. Any $p$ dividing $1 + n + \cdots + n^{q-1}$ satisfies $p \equiv 1 (\mod q)$, whenever $ n \not \equiv ...
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1answer
58 views

Direct limit of subgroups

Let $G$ be a group and $G^i$ a collection of subgroups which form a direct system over a directed set $I$, so $i\leq j \iff \exists\; \varphi^i_j: G^i\to G^j$ where $\varphi^i_j$ is the inclusion map. ...
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1answer
74 views

The Trivial Centre of Group $G/Z(G)$ [closed]

Let $G$ be a group such that $G = G'$, where $G'= [G,G]$ is the Derived Subgroup of $G$. Prove that the centre o group $G/Z(G)$ is trivial, that is, prove that $Z\left(G/Z(G) \right) = 1$.
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Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
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1answer
83 views

Presentation of a group: Show that $\langle a|a^2\rangle =\{1,a\}$.

Example: $\langle a|a^2\rangle=\{1,a\}$. After reading the definition of presentation of a group, I find myself cannot understand the above example given. I don't know which part of the definition I ...
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1answer
35 views

Short exact sequence with binary tetrahedral group does not split

The following is a short exact sequence, where $T$ is the binary tetrahedral group (equivalently the Hurwitz units), and $Q$ is the quotient of $T$ by $\mathbb{Z}/2$. $1 \rightarrow \mathbb{Z}/2 ...
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2answers
94 views

Group theory and group representation

I am fairly new to group theory and representation. I am currently looking at faithful representations. I am not quite sure what is the "use" of a faithful representation. I cannot find any "easy to ...
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1answer
67 views

Find the Automotphism group of direct product of Z(mod m) & Z(mod n).

Find the Automotphism group of direct product of $\mathbb{Z}(\operatorname{mod} m)$ and $\mathbb{Z}(\operatorname{mod} n)$. We know Automorphism of direct product of $\mathbb{Z}(\operatorname{mod} ...
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1answer
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Why the paired orbit has the same size here?

enter link description here On this proof, he just showed that the the paired orbit of (a,b) has the same size, but this seem has nothing to do with the size of corresponding paired orbit of an orbit ...
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Number of Sylow 2-subgroups of a special linear group

Find the number of Sylow $2$-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo $3$). I think it will be $1$. But I failed to prove it using the counting principle. It has $4$ ...
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1answer
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Representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely ...
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64 views

Number of actions of $\mathbb Z$

Let $X$ be a finite set. Determine the number of actions of $\mathbb Z$ on $X$. If $X$ is a finite set with $|X|=m$, then $|\{f:X \to X : \text{f is bijective}\}|=m!$. Finding the number of actions ...
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225 views

Infinite group having composition series

Example of an infinite group having composition series. I have examples as infinite alternating group and projective special linear group. But I want example other than infinite simple group.
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62 views

A question on the order of an element involving relatively primes

This question is based on an exercise that comes from the second chapter of Malik's Fundamentals of abstract algebra which states as follows (I paraphrase): Let $(G, *)$ be a group and $x\in G$. ...
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1answer
67 views

What can you say about the order of this element?

I am self-studying algebra and encountered the following problem: If $b^{-1}ab=a^2$ and $c^{-1}ac=a^3$ and $b,c$ has orders $4$ and $3$ respectively, what can you say about the order of $a$? In ...
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every Abelian group is a converse lagrange theorem group

Let $G$ be a finite abelian group, then $G$ has a subgroup of order $n$ if and only if $n\mid G$. Proof: by Lagrange if $H\leq G$ then $|H|$ divides $|G|$ so this proves one of the implications. We ...
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106 views

The smallest composite integer $n$ such that there is a unique group of order $n$?

We need to find the smallest composite integer $n$ such that there is a unique group of order $n$. Attempt: Let us suppose that $n= ab$ is a composite integer where $a,b$ are integers such that ...
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1answer
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$S=\{a\in G:f(x)=f(ax)\; \forall x\in G\}$ is a subgroup of $G$

Let $G$ be a group and $f:G\rightarrow G$ a function. Let $S=\{a\in G:f(x)=f(ax)\; \forall x\in G\}$. Prove that $S$ is a subgroup of $G$. This is my first encounter with functions in this ...
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1answer
65 views

Is there a concise argument for why this group operation is associative?

Given disjoint groups $(G,\cdot)$ and $(H,\ast)$ and an isomorphism $f:G\to H$, I've been able to show that the binary operation $\diamond$ on $G\cup H$ defined by $a\diamond b = a\cdot b$ if $a,b ...