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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114 views

Number of left cosets of the special linear group in the general linear group

Let $F$ be a field of cardinality $q$. I need to prove that $\frac{\left |{GL_n(F)}\right |}{\left |{SL_n(F)}\right |}=q-1$. I try to find a bijection between the left cosets of ${\left |{SL_n(F)}\...
2
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1answer
808 views

Subgroups and quotient groups of finite abelian $p$-groups

Motivation The fundamental theorem of finite abelian groups gives us a concise description of the isomorphism types of finite abelian $p$-groups $G$ (in the following, $p$ is a fixed prime). The ...
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1answer
50 views

Are these equivalent definitions of a quotient group?

After grappling with the concept of a quotient group I have come across two different definitions of a quotient group. The first which is used in Algebra Chapter 0 uses an equivalence relation $\sim$ ...
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1answer
48 views

Show that inversion map is a bijection $G/H \to H\! \setminus \! G$, so that number of left cosets always equals the number of right cosets

Show that the inversion map is a bijection $G/H \to H\! \setminus \! G$, so that the number of left cosets always equals the number of right cosets (even if $G$ is infinite). Here $G/H$ is the ...
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2answers
91 views

If $X$ and $Y$ are g-equivariant homeomorphic then $X/G$ and $Y/G$ are homeomorphic

Let $X$ and $Y$ be $G$-sets (That is the group $G$ acts on $X$ and $Y$). We say that the function $f: X\to Y$ is G-equivariant if $f(g.x) = g.f(x)$ for all $x\in X$ and all $g\in G$. Prove that if $X$...
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1answer
30 views

If $G \cong H \times \mathbb{Z}_2$, show that $G$ contains an element $a$ of order $2$ with the property that $ag = ga$ for all $g \in G$.

I have so far that there is one element in $H \times \mathbb{Z}_2$ with order two which commutes with everything else in that group, namely $(e_H, 1)$. This is presuming that $\mathbb{Z}_2$ is a group ...
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0answers
42 views

Can $G$ be a union of some of$H$ copies?

Let $G$ be a finite group and $H$ be a proper subgroup. Then we know that $$G\neq \bigcup_{g\in G} H^g$$ I wonder whether this is possible $$G= \bigcup_{\sigma \in Aut(G)} \sigma (H)$$ If there ...
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1answer
19 views

relations in groups with 2 generators

In general, given a group $G=<x,y \vert x^n=y^m=1, yx=x^iy>$, how do we re-express $y^ax^b$ as $x^cy^d$ where $c,d$ in terms of $n,m,i,a,b$? (Currently I am attempting questions where $i=2$. I ...
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1answer
80 views

If $G$ is a group of order $48$, show that the intersection of any two distinct Sylow $2$-subgroups has order $8$

All I know is that we have $3$ Sylow-$2$ subgroups of order $16$. $$o(H \cap K)= o(H)o(K)/o(HK)$$ How to proceed further?
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1answer
83 views

Order of the center of a group - odd?

I was just thinking, does the order of the center of a group have to be odd? If not, then there is definitely something wrong with my reasoning in the following proof. Suppose the center of a group $...
2
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1answer
61 views

If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?

Given $|G|=p^2$ then how can you deduce $G\cong C_{p^2}$ or $G \cong C_p \times C_p$ I have shown that G is abelian, not sure what to do next
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1answer
59 views

solve the exercise in an alternative way

I have this exercise: Assume that G and H are finite groups and that $|G|$ and $|H|$ are relatively prime. Show that the only group-homomorphism $\phi: G \rightarrow H$ is the trivial one. I want to ...
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0answers
47 views

Orbits of $2 \times 2$ matrices over $\Bbb F_2$

Let $X$ be the set of $2 \times 2$ matrices over field $\Bbb F_2$, and $G \subset X$ be the group of invertible $2 \times 2$ matrices over $\Bbb F_2$ Let $G$ act on $X$ by $g*x = gxg^{-1}$ Need to ...
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2answers
236 views

Parity of permutation example

I know the definition of parity of permutation. But what does that look like in examples? For example, if the number of permutations is odd, then the sign of permutation in $-1$. What does this mean? ...
3
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3answers
1k views

Number of non-isomorphic non-abelian groups of order 10

Number of non-isomorphic non-abelian groups of order 10 Let $G$ be a group of order 10.Let $a\neq e \in G$ then it is not possible that all elements are of order 2 otherwise $G$ becomes abelian. ...
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1answer
174 views

Suppose $G$ is a group with $|G|=35$. Prove that if $H$ is a subgroup of $G$ with order 7, then $H$ is a normal subgroup of $G.$

(1) Suppose $G$ is a group with $|G|=35$. Prove that if $H$ is a subgroup of $G$ with order $7$, then $H$ is a normal subgroup of $G.$ (2) Suppose that G is a group with $|G|=35.$ Prove that if $G$ ...
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0answers
87 views

Integrating over a symmetric-group function (elements being permutations)

I would like to integrate a permutation of a function. Namely I have the following: $\sum_{\sigma, \sigma'\in S_{n+1}}\int_{-A}^A dz_1dz_2 ... dz_{n+1} \left(z_{\sigma(n+1)}e^{i\alpha\sum_{j=1}^{n+...
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5answers
266 views

Confused about the group of permutations $S_{n}$

In an exercise, I must prove $S_{n}$ is generated by 2 elements. I'll ignore here the trivial case $n = 1$. Let $I_{n} = \{1, 2, 3, ..., n\}$. I then defined $f : I_{n} \rightarrow I_{n}$ by $f(1) = 2$...
2
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1answer
87 views

How unitarize an irreducible representation of a finite group?

Let $G$ be a finite group acting irreducibly on the space $V$. $$\psi : G \to Aut(V)$$ Question: How unitarize the representation $V$? I'm looking for a computable process.
2
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1answer
104 views

do we have any special algorithms or software to find all 2-sylow subgroups of a group?

I am working on a project that involves 2-sylow subgroups of groups,one thing that I need to do is to find all 2-sylow subgroups of a group and check that it is cyclic or not, now my question is that ...
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1answer
43 views

If $S$ is an Infinite Set, and $f(S)\subseteq{G}$ is a Set that Generates a Group $G$, then the Cardinality of $G$ is $\le$ Cardinality of $S$?

As the header says, if we map an infinite set $S$ into $G$ by $f$, such that $f(S)$ generates $G$, then is $|G|\le|S|$? I know that this should be true in the case where $S$ does not include into $G$. ...
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3answers
327 views

Existence of subgroup of order six in $A_4$

Show that the alternating group $A_4$ of all even permutations of $S_4$ does not contain a subgroup of order $6$. For me am thinking to write all elements of $A_4$ and trying to find every ...
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2answers
188 views

The smallest non-abelian group $G$ with a non-normal subgroup [closed]

This time I need to find the smallest non-abelian group $G$ with a non-normal subgroup, then my questions are: 1)Can someone help me to find it? 2)Once we find it, How Can you prove that it is the ...
4
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1answer
133 views

About direct limit of groups

Let $G_i$ be sequence of groups for $i\in \mathbb N$ and Let $\phi_i$ be a monomorphism from $G_i$ to $G_{i+1}$. Let $\Sigma$ be the direcet limits of $G_i$ under the embeddings of $\phi_i$. Let $\...
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1answer
36 views

Given a group is finite and non-abelian, why is the left coset with the centre of the group non-cyclic?

Assume $T$ is finite and non-abelian then why is $T/Z(T)$ non-cyclic? Where $Z(T)$ is the centre of the group $T$. I've shown $Z(T)$ is a normal subgroup of T, but not sure what to do next or if ...
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1answer
252 views

Computing the order, inverse, and parity of a permutation

How do you compute the order, inverse and parity of $\alpha=(12)(43)(13542)(15)(13)(23)$? Please explain all steps taken to get the answer. I guess my thought process was to first put it into a ...
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2answers
44 views

Way to show that there exists $n\neq 0$ such that $n^{(p-1)/2}\neq 1 \mod p$

Suppose that $p\ge3$ is a prime and that $0\neq n \in \mathbb{Z}/p\mathbb{Z}$. Suppose also that we want to show that there is an $n$ such that $n^{(p-1)/2} \neq 1 \mod p$. One way to show this would ...
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1answer
51 views

Coset to a power

my book states that if $N \trianglelefteq G$ then $(gN)^\alpha = g^\alpha N$ for $\alpha \in \mathbb{Z}$ The proof should be simple by induction but I can't understand how $(gN)^0 = [N = g^0N]$. How ...
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2answers
75 views

Every subgroup of $(\mathbb {Z_n},+)$ is closed under multiplication

I am stuck in this proof that every subgroup of $(\mathbb {Z_n},+)$ is also a subring. which requires me to prove it is closed under multiplication. I have to show if $a,b \in G<\mathbb {Z_n}$ , ...
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2answers
31 views

Homomorphisms $h:C_5 \to Aut(C_{31})$ (first course in group theory)

I am asked to show explicitly all homomorphisms $h: C_q \to Aut(C_{31})$ for $q=5,3,2,29$ For $q=5$, $C_5 =\{1,y^2,...,y^4\}$ and $Aut(C_{31})=C_{30}=\{1,\alpha,...,\alpha^{29}\}$ so we know the ...
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1answer
74 views

Elements of the field $F_2[x] / (x^3 + x + 1)$

What do elements of the field $F_2[x] / (x^3 + x + 1)$ look like? I know this is isomorphic to $F_8$, and that its elements have max degree of 2, so that leaves me with $0$, $1$, $x$, $x^2$, $x+1$ , $...
1
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1answer
46 views

Intersection of conjugate subgroups with infinite index.

Is there a group $G$ with a subgroup $H\subseteq G$ of finite index and an element $g\in G$ such that $H\cap g^{-1} H g$ has infinite index in $G$?
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0answers
48 views

Characterisation of subgroups of finite index of $SL_2(\mathbb Z)$

Is there a complete characterisation of subgroups of finite index of $SL_2(\mathbb Z)$?
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2answers
72 views

Why can't $A_4$ has a subgroup of order $6$? [duplicate]

Can anyone provide with an explanation of why the group $A_4$, which is the group formed by the set of even permutations of $S_4$ under the operation of composition of functions, can not have an order ...
5
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1answer
70 views

Every group as full symmetry group of points in $\mathbb R^d$

Does every finite group $G$ have the property that it is isomorphic to a full symmetry group of some set of points in $\mathbb R^n$ for some $n$
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0answers
58 views

How many conjugates does a regular permutation group have?

Let $G$ be a finite group of order $n$. Consider $G$ as a regular subgroup of the symmetric group $S_n$. What is the number of conjugate subgroups to $G$?
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1answer
40 views

Is H a subgroup of G?

Let K be a subgroup of a group G and H be a subgroup of K. Is it true that H is a subgroup of G? Justify. It is obvious that H is a subgroup (like subset of a subset is a subset of the original set). ...
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2answers
35 views

About the statement $g_1 \sim g_2 \iff Hg_1=Hg_2$.

I have a question about my notes of algebraic structures. Let $G$ be a group. Let $H \leq G$. We define that \begin{equation} g_1 \sim g_2 \iff g_1g_2^{-1} \in H. \end{equation} My professor ...
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1answer
44 views

Question about a proof regarding cosets.

We proved this lemma in my class Let $G$ be a group. Let $H\leq G$. Then \begin{equation} \phi:(G/{\sim})=G/H \to H\backslash G =(G/{\approx}), \end{equation} with $\phi(Hg)=g^{-1}H$, is ...
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1answer
74 views

Permutation(?) mapping [duplicate]

Problem Statement: Let $G$ be a finite group, say a group with $n$ elements, and let $S$ be a nonempty subset of $G$. Suppose $e \in S$, and that $S$ is closed with respect to multiplication. ...
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2answers
446 views

Is a subgroup of a topological group a topological group?

I'm trying to solve the problem from Munkres: Let $H$ be a subspace of $G$. (Where $G$ is a topological group). Show that if $H$ is a subgroup of $G$, then both H and H closure are topological groups. ...
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0answers
41 views

Coset state for non-abelian hidden subgroup problem

On page 14 of his classic paper, Quantum factoring, discrete logarithms and the hidden subgroup problem, Jozsa introduced a function $f$ for the non-abelian hidden subgroup representation of the graph ...
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2answers
133 views

Order of elements in Z/10Z

Can anyone help me with the following question: How do I find the elements of order 5 i the group $\mathbb{Z}/10 \mathbb{Z}$?
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2answers
80 views

Prove that the set of all periods of a function is a subgroup

Let $G$ be a group and $f: G \to G$ a function. A period of $f$ is any element $a \in G$ such that $f(x) = f(ax)$ for every $x \in G$. Prove that the set of all periods of $f$ is a subgroup of $G$. ...
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1answer
53 views

Can you check my proof that $H$ is characteristic? [duplicate]

Suppose $|G|=pm$,where $p\nmid m$. Then if $H\mathrel{\unlhd}G$ and $|H|=p$, than $H$ has to be characteristic. We call a subgroup characteristic when $\varphi(T)\subset T$, $\forall \varphi \in Aut(G)...
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1answer
121 views

An Example for a Graph with the Quaternion Group as Automorphism Group

I am reading "Graphs of Degree Three with given Abstract Group" (by Robert Frucht) where the author describes (somewhat tedious) algorithms to construct suitable graphs starting from a given group. I ...
0
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1answer
48 views

Homomorphism and images

G is a finite group, $\phi:G \to G$ a homomorphism. $\psi:G \to G$ is a homomorphism defined by $\psi(x)=\phi(\phi(x))$. Prove that $(\ker\phi= \ker \psi)\implies($Im$ \psi=$Im$ \phi)$. Can someone ...
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1answer
40 views

Let A be a simple subgroup of a group G and N be a normal subgroup of G. Show that the only normal subgroups of AN containing N, are N and AN.

By definition, since $A$ is simple, it contains only itself and ${e}$. If $A$ only contains itself then a subgroup $A$ of $AN$ cannot contain $N$ because there are no common elements between $A$ and $...
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0answers
60 views

Structure of the semidirect product decomposition

I'm looking at a complicated group that involves many semidirect products, and I realized that I have a fundamental confusion about how to use the structure of a semidirect product decomposition of a ...
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2answers
47 views

Let H and N be normal subgroups of a group G with H $\cap$ N = {e}. Prove that hn = nh for all h $\in$ H and n $\in$ N.

What does H $\cap$ N = {e} imply here? Is it related to the divisibility of the orders of H and N? If so, how do I relate that to show commutativity? Thanks in advance.