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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$H,N(H)$ are subgroups of $G$ show that $H\lhd N(H)$

Let $G$ be a group and $H$ subgroup of $G$, $N(H):=\{g\in G; gHg^{-1}=H\}$ $N(H)$ is also subgroup of $G$. I need to prove that $H$ is a normal subrgoup in $N(H)$ Attempt: $H\lhd N(H) \iff ...
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30 views

A singleton set $\{g\}$ can be regarded as a unary relation in $G$. Why?

Theorem 1.1. A relation $R \subseteq M^n$ is definable if and only if every automorphism of every elementary extension of $M$ preserves $R$. For a proof, the reader can see [4]. Suppose we ...
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1answer
24 views

'Large' closed subgroup

I am working through a paper in the field of differential geometry (Yang-Mills theory) and the author writes: 'We assume the Riemannian manifold $(M,h)$ admits a large closed subgroup $K$ of the ...
2
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1answer
25 views

The abelian group of smallest order and smallest non prime integer n divides |G| but G doesn't have an element of order n?

I don't know how to think of an example. What's an example of such a group. It doesn't make sense to me because if it is a finite abelian group, it can be written as a direct product of the integers ...
2
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3answers
52 views

How is $\lbrace a_1, a_2, …, a_n : a_i \in \Bbb Z_2\rbrace$ a group?

I was asked to prove that if we define \begin{equation*} \Bbb Z_2^n = \lbrace a_1, a_2, ..., a_n : a_i \in \Bbb Z_2\rbrace \end{equation*} then it's a group under the operation of addition like ...
2
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2answers
26 views

Show that $N(H):=\{g\in G; gHg^{-1}=H\}$ is subgroup of $G$

Let $G$ be a group and $H$ subgroup of $G$, $N(H):=\{g\in G; gHg^{-1}=H\}$ I need to prove that $N(H)$ is subgroup of $G$. It's almost the same question like :$\forall g \in G, gHg^{-1} = H ...
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0answers
31 views

Let $H$ is a subgroup of $G$. When $G=HZ(G)$? [on hold]

Let $H$ is a subgroup of $G$. found conditions that $G=HZ(G)$?
1
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1answer
21 views

Prove this result relating to the sign of a permutation

Suppose that $\phi \in S_n$ is a permutation. Suppose also that $\psi = \phi \circ (i,j),$ where $1 \leq i, j \leq n.$ Why does it follow that sign$(\phi) = $ $-$sign$(\psi)$?
2
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1answer
69 views

Trirectangular tetrahedron

Looking at http://mathworld.wolfram.com/TrirectangularTetrahedron.html I wonder what the symmetry group of a trirectangular tetrahedron is?
1
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1answer
26 views

Is this proof correct for $p_1\times p_2 \mod k = p_3$?

I am trying to prove that if $\gcd(p_1, k) = \gcd(p_2, k) = 1$, then $$\gcd(r, k) = 1$$ where $r = p_1\times p_2 \mod k$. This fact is essential to guarantee that a unit group $U(n)$ of a group ...
2
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3answers
88 views

Verifying if a multiplication table is from a group

I'm asked to verify which of these multiplication tables form a group. I'm having problems to see which of the axioms for a group are violated in each table. In (a), I couldn't find an element $e$ ...
1
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1answer
41 views

covering subsets

Let $A=\left\{ {1, 2, \ldots, n}\right\}$. Let $B$ be the set of all size $m$ subsets of $A$. $B=\left\{{B_1,B_2, \ldots , B_{\binom{n}{m}} } \right\}$, $ |B_i|=m$ then we want to find $k$ subsets ...
4
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2answers
87 views

Structure of the group $\{1+p\mathbb Z_p \}$

In preparation for algebraic number theory I am reading Serre : A course in Arithmetic. I stuck in understanding a proof (p.17): Notation: $U_n=1+p^n\mathbb Z_p$ Actually there are many things ...
0
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1answer
24 views

prove that if $N\lhd G$, $ M\lhd G$, $M\bigcap N=\{e\}$ so: $mn=nm , \forall n \in N,\forall m\in M$ [duplicate]

prove that if $N\lhd G$, $ M\lhd G$, $M\bigcap N=\{e\}$ so: $mn=nm , \forall n \in N,\forall m\in M$
3
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1answer
22 views

Product of any two non disjoint cycles

Suppose you have a permutation $\sigma = \sigma_1 \sigma_2$ where $\sigma_1 = (i_1 i_2...i_k)$, $\sigma_2 = (j_1 j_2...j_l)$ and $i_{k_1},i_{k_2},...,i_{k_r}$ are equal to ...
0
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2answers
45 views

Is it possible for an element of a multiplicative group to have undefined order?

This might be a stupid question but here it is: Let $(G, \phi)$, where $\phi :G \times G \rightarrow G$, be a multiplicative group with identity element $e$. Then is it possible that $\exists a \in ...
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0answers
18 views

Equivalence of irreducible representations of special linear group SL$(n)$ via those of GL$(n)$ and invariant total anti-symmetric tensor

I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. Theorem 13.14 discusses the the equivalence of irreducible representations of special linear group ...
4
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0answers
75 views

Aut(G) is abelian

I've heard of this (open?) problem: Classify groups G such that Aut(G) is abelian. What I discovered: Any characteristic abelian subgroup is cyclic. Center is cyclic. Commutators are cyclic. ...
3
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1answer
30 views

$a,b,N$ are integers. Prove $x=x_0+\cdots$, $\ \ y=y_0+\cdots $ are solutions to $ax+by=N$

I'm asked to prove that if $a,b,N$ are integers, then in the equation: $$ax+by=N$$ I must prove that the integers $$x=x_0+\frac{b}{d}t,\ y=y_0-\frac{a}{d}t$$ are solutions to the equation. where ...
26
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3answers
1k views

If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ ...
4
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1answer
118 views

What is mathematical structure?

When we have an isomorphism, between 2 groups or vector spaces let us say, then it is said to be structure preserving. An isomorphism exists when there is at least one mutually invertible morphism ...
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1answer
18 views

Classifying the central product HK of two cyclic groups [on hold]

Let group $H$ be a direct product of cyclic groups $C_1$ and $C_2$ of order $p$ and $p^2$ respectively. Let $D=\{x\in H\mid \text{ord}(x)\leq p \}$. D is generated by $C_1$ and subgroup $E$ of $C_2$ ...
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0answers
15 views

Finding efficiently finite groups whose set of commutators is not a subgroup

The set of commutators of a group might not be a subgroup of the group. I give here such an example from P.J. Cassidy. It is an infinite group. It is possible to derive from this example one of a ...
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1answer
26 views

all abelian groups with 625 elements with 24 elements of order 5

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form $M = R/(p^{e_1}) \oplus \cdots \oplus R/(p^{e_t})$. Let $_pM = \{m \in M: p ...
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2answers
63 views

The center of a group $G$ is a subgroup of $G$ [duplicate]

Definition for the center of a Group: The center $Z(G)$ of a group $G$ is the subset of elements in $G$ that commute with every other element of $G$. Theorem: The center of a group $G$ is a ...
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votes
0answers
38 views

I want to know if the below sentence is true and why?

I want to know if the below sentence is true and why? Let $G$ be an insoluble finite group then there exists $\pi\subset\pi(G)$ such that if $K=O_{\pi}(G)$ and $\bar{G}=G/K$ it follows that ...
6
votes
5answers
139 views

An example of a group such that $G \cong G \times G$

I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group?
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1answer
26 views

Examine if $\phi$ is a homomorphism and determin $\ker \phi$ and $Im \phi$.

Let $G=(\mathbb C^*,\cdot), G'=(\mathbb R^*,\cdot)$ and $\phi : G\to G'$ be defined by $\phi(z)=|z|, z\in \mathbf C^*$, where $\mathbf C^*=\mathbf C-\{0\}$ and $\mathbf R^+$ is a set of all positive ...
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5answers
53 views

If $A$ and $B$ are subgroups of a group $G$, is their product also a subgroup of $G$?

Is there a necessary and sufficient condition for $AB = \{ab|\;a\in A, \; b\in B\}$ to be a subgroup of a group $G$, given $A, B$ are subgroups of $G$?
2
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1answer
36 views

Characters of transitive finite permutation group

I know that Frobenius reciprocity helps us to solve this problem, but I don't know why: Let $ G $ be a transitive finite permutation group with permutation character $ \pi $. If $\chi $ is an ...
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1answer
16 views

Two-step subgroup test “IFF” condition

The theorem for the two-step subgroup test says: The subset H of a group G is a subgroup IFF the binary operation of 2 ordered pairs of elements of H are in G and for each element in H, there each ...
1
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1answer
28 views

One-to-ones-ness of Group Elements

I came along this theorem in a book, saying that if $a,b,c \in G$, where $G$ is a group, then if $ab = ac$, then $b=c$. It looks like if we assume that all elements in the set are functions (because, ...
4
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0answers
23 views

What is $Hom((S^1)^k , (S^1)^n)$?

I am trying to find $Hom_{gp}((S^1)^k , (S^1)^n)$ , which is the set of continuous group homomorphisms from the $k$ dimensional torus to the $n$ dimensional torus where $1 \leqslant k \leqslant n$. ...
2
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2answers
39 views

Groups - Proof that $(ab)^{-1} = b^{-1}a^{-1}$

I read this proof in a book: Prove that if $a, b \in G$, then $(ab)^{-1} = b^{-1}a^{-1}$. Proof: Let $a,b \in G$. Then $abb^{-1}a^{-1} = aea^{-1} = aa^{-1} = e$. Also, $b^{-1}a^{-1}ab = e$. ...
3
votes
3answers
55 views

$G$ a group and $H$,$K$ subgroups, $kHk^{-1} \subseteq H \implies kHk^{-1} = H$?

As post said, if $G$ a group and $H,K \leq G$ and for FIXED $k \in K$ does $kHk^{-1} \subseteq H$ imply that $kHk^{-1} = H$ ?
10
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0answers
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Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g: \mathbb R \to \mathbb R$ be the permutations defined by $f: x \mapsto x+1$ and $g: x \mapsto x^3$, or maybe even have $g:x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la ...
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votes
0answers
18 views

If $ G $ is finite group and $ K $ is nilpotent subgroup of $ G $. [on hold]

If $ G $ is finite group and $ K $ is nilpotent subgroup of $ G $. is there theorem that said can let $ K = K_{\pi^{\prime}}K_{\pi^{\prime}}$ that $K_{\pi}$ is $\pi$-Hall subgroup of $ K $ and $ ...
0
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2answers
45 views

Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$? [duplicate]

Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$?
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2answers
92 views

Elementary question in Group Theory with less prerequisite

Here I am posing a problem, which my beginning students of algebra were discussing for long time. Question: Without using theorem of Cauchy or Sylow, can we show that a group of order $15$ contains ...
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0answers
31 views

Is group theory a generalization of number theory [on hold]

The applications of group theory are abound. Many mathematical objects are examined by associating groups to them and studying the properties of the corresponding groups. But number theory and Graph ...
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1answer
40 views

For any finite group $G$ and for any natural number $n$, does there exist a group $H$ such that $|H|=n|G|$ and $G$ is a normal subgroup of $H$? [on hold]

For any finite group $G$ and for any natural number $n$, does there exist a group $H$ such that $\left\vert H\right\vert=n\left\vert G\right\vert$ and G is a normal subgroup of H?
4
votes
3answers
51 views

How to prove that $(G,*)$ is a group?

Let $G=\mathbb{R_0}\times\mathbb{R}$ where $\mathbb{R_0}=\mathbb{R}\setminus\{{0}\}$. Define operation $*$ on $G$ by $(a,b)*(x,y)=(ax,a^2y+b)$. I'd like to prove that $(G,*)$ is a group. ...
0
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1answer
35 views

uniqueness of identity of a group G

The theorem for uniqueness of identity of a group says there is one identity element $e$ in a group and this element $e$ is unique. My book states the proof as follows: $a.e=a$ for all $a \in G$ and ...
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2answers
32 views

How to prove that $a^{|G|}=e$ if a $\in G $

How to prove that; $a^{|G|}=e$ if a $\in G $ if $G$ is a finite group and $e$ is its identity. I think this could be done through pigeonhole principle but I don't want to use the Lagrange ...
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1answer
26 views

General question on notations when dealing multiplicative and additive modulo

One of the property for the requirement for a set to be a group is associativity. Under ordinary multiplication: $\large{a(bc)=(ab)c}$ Under ordinary addition: $\large{a+(b+c)=(a+b)+c}$ What ...
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0answers
19 views

Is $\ker(nat_{H})=H$ a true statement? [on hold]

Is $\ker(nat_{H})=H$? $nat_{H}$ defines as $nat_{H}(a)=a*H$ I know what $ker$ and $nat_{H}$ are but I am not familiar with $\ker(nat_{H})$.
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1answer
48 views

Frobenius Reciprocity and a character theory problem

How Frobenius Reciprocity can help us to solve these two problems: Let $ H $ be a subgroup with index $ m $ in the finite group $ G $. Let $ F $ be an algebraic closed field of characteristic $ 0 $. ...
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2answers
106 views

Is every group isomorphic to some nontrivial quotient group?

For any group $G$, does there exist a group $H$ and a nontrivial normal subgroup $N$ of $H$ such that $H/N\cong G$?
0
votes
1answer
24 views

Do there exist nontrivial quotient groups of arbitrary finite order?

For any $n\in \mathbb{N}$, does there exist a group $G$ and a nontrivial normal subgroup $N$ of $G$ such that $\left\vert G/N\right\vert =n$?
0
votes
2answers
49 views

Is the set $SL(2, \mathbb F)$ an Abelian group?

For the set $SL(2,\mathbb F)$, where $\mathbb F$ are entries from either $$\mathbb{Q},\mathbb{R},\mathbb{C} \text{ or } \mathbb{Z}_p \text{ (p is prime)}$$ How should I start by checking this matrix ...