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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Prove that if $H$ is a unique subgroup of order $|H|$ then $H$ is a characteristic subgroup.

Let $G$ be a group and $H\leq G$. $H$ is a unique subrgroup of order $|H|$. Prove that $H$ is a characteristic subgroup. I tried to show by contradiction that if $\phi (h)\not \in H$ then I can find ...
4
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0answers
51 views

condition for a group to be abelian [duplicate]

I’d like to prove: Let $G$ be a group and $m,n$ two relatively prime numbers. If $x^my^m=y^mx^m$ and $x^ny^n=y^nx^n$ for all $x,y\in G$, then $G$ is abelian. Thanks for help in advance.
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2answers
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$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
4
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3answers
127 views

Every normal subgroup is the kernel of some homomorphism

Clearly the kernel of a group homomorphism is normal, but I often hear my professor mention that any normal subgroup is the kernel of some homomorphism. This feels correct but isn't entirely obvious ...
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2answers
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Subgroups of $S_n$ with exactly one fixed point for each element all have the same fixed point.

Let $G$ be a subgroup of $S_n$ (where $n$ is a positive integer) such that each non identity element $g\in G$ has exactly one fixed point. Prove there is an element of $[n]$ that is fixed by every ...
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1answer
50 views

True or false simple algebra questions (centralizers, conjugacy classes, normal groups, abelian groups) [closed]

Can someone please verify my answers to the following questions? Note: This is NOT homework! Answer true or false to the following questions: Two elements of a group in the same conjugacy ...
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2answers
28 views

Permutations with Condition

I have looked at this old problem in my textbook: How many permutations $\pi \in S_n (n \geq 3)$ meet the requirement: $\pi (1) < \pi (2) $ or $\pi (1) < \pi (3)$? I am not sure how to ...
2
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2answers
40 views

Equations in groups

I want to solve an equation $$f(\sigma , \tau , \delta)=1$$ where $\sigma,\tau,\delta$ are elements from a given group $G$, and $1 \in G$ is the unit element. When I say solve I mean give sufficient ...
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0answers
22 views

$ G $ is satisfies the maximal permutizer condition if $ G $ is supersoluble?

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
19
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3answers
399 views

Stanford math qual: Abelian groups $G$ satisfying $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to 0$

I am studying for my qualifying exams and came across the following question: Find all abelian groups $G$ that fit into an exact sequence $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to ...
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1answer
26 views

Minimal order of a group with a particular property

I fix an integer $n$. I am looking for a group $G$ for which there exist elements $g_1, \dots, g_n \in G$ and $h_1, \dots, h_n \in G$ such that $$ h_kg_k^{-1} \neq h_j g_i^{-1}$$ as long as ...
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vote
1answer
52 views

Can two non-abelian groups have an abelian product or coproduct?

It is easy to see that two or more abelian groups must have an abelian product (and coproduct, since these constructions coincide in $\sf Ab$.) I'm not sure how to proceed with this; I just thought ...
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2answers
29 views

$H\subseteq G$, $N\triangleleft G$, and showing $|[G:N]|$ is a prime number

Let $G$ be a finite group and let $N\triangleleft G$ a normal subgroup. It is given that if $H\subseteq G$ such that $N\subseteq H\subseteq G$, then $H=N$ or $H=G$. Show that $|[G:N]|$ is a prime ...
6
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1answer
75 views

Does anyone know the Burnside Matrices?

For $G$ a fine group with conjugacy classes $C_1,\dots C_k$ we introduced the Burnside Matrices $A_r$ where $1<r<k$ with entries: $$A_r := \Big(\sqrt{\frac{|C_t|}{|C_s|}}a_{rst}\Big)_{1\leq ...
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0answers
27 views

Is there any “Brunnian-like” braid which is not a pure braid in $B_n$ with $n\geq 3$?

A braid $\beta$ is called Brunnian if it satisfies $\beta$ is a pure braid; $\beta$ becomes a trivial braid after removing any of its strands. Obviously in $B_2$, every braid satisfies condition ...
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0answers
37 views

For which integers $r$ is $\sigma ^r$ also a $k$-cycle? [duplicate]

Let $\sigma$ be a $k$-cycle in $S_n$. For which integers $r$, is $\sigma ^r$ also a $k$-cycle? I think I managed to prove that this is true iff $(k,r)=1$, but my proof was too long and not elegant ...
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0answers
67 views

Orbit of a Weight Vector?

Given some element $\phi$ of a representation $R$ of a group $G$, the orbit $G(\phi)$ of $\phi$ is defined as the set $g \phi \ \forall \ g \in G$. We can write every element of a given ...
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Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
2
votes
1answer
92 views

Rank 3 permutation groups

Let $G \leq Sym(\Omega)$ be a finite permutation group of rank 3, $\alpha \in \Omega$ and $g,h \in G$ such that $x_1 := g(\alpha)$ and $x_2 := h(\alpha)$ are not equal. Now my question is: Is there ...
5
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1answer
66 views

Haar measure on SO(n)

I am interested in describing the group of special orthogonal matrices SO(n) by a set of parameters, in any dimension. I would also like to obtain an expression of the density of the Haar measure in ...
0
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1answer
21 views

Prove that if $a\in G^n$ and $g\in G$ then $g^{-1}ag\in G^n$

Let $G$ be a finite group and $n\in \mathbb{N}$. For all $a,b\in G$ there is: $(ab)^n=a^nb^n$. Define $G^n=\{g^n\ |\ g\in G\}$. Prove that $G^n$ is a subgroup of $G$ and that if $a\in G^n$ and ...
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1answer
44 views

Let $G$ be finite group. if $A,B\le G$ with orders $4, 5$ respectively then $A \cap B$? [closed]

Let $G$ be finite group. If $A$ and $B$ are subgroups of $G$ with orders 4 and 5 respectively, what is $A \cap B$ ?
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1answer
42 views

Representations of the form $\varphi: G \rightarrow GL(V)$ vs $\phi: G \rightarrow Aut(A)$

Standard representation theory studies homomorphisms of the form $\varphi: G \rightarrow GL(V)$ where $V$ is a vector space. How much does the focus of representation theory change if one considers ...
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2answers
26 views

Group acting on $X$ and element of normal subgroup $H$ fixes an element of $X$ implies $H$ fixes all of $X$

A group $G$ acts on a set $X$ transitively and a normal subgroup $H$ fixes a point $x_{0} \in X$, i.e. $h \cdot x_{0}=x_{0}$ for all $h \in H$. Show that $h \cdot x = x$ for all $h \in H$ and $x \in ...
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2answers
66 views

Number of cycles in complete graph

How many number of cycles are there in a complete graph? Is there any relation to Symmetric group?
5
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2answers
76 views

$G=\langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$ is torsion-free.

I have to prove that $G=\langle x,y\ |\ x^{-1}y^2x=y^{-2}, y^{-1}x^2y=x^{-2} \rangle$ is torsion-free. Some things about this group that I understand are first show that $M=\langle (xy)^2,x^2,y^2 ...
2
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4answers
83 views

Subgroup of $\mathbb{Z}$ generated by two positive integers

An exercise from Aluffi's Algebra book. Let $m,n$ be positive integers and consider the subgroup $\langle m,n\rangle$ of $\mathbb{Z}$ they generate. As a subgroup of $\mathbb{Z}$ it will be equal ...
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4answers
161 views

Find the order of an element of finite group

Let $G$ be a finite group and $g,h\in G-\{1\}$ such that $g^{-1}hg=h^2$. In addition $o(g)=5$ and $o(h)$ is an odd integer. Find $o(h)$. I know from a previous exercise that if there exists a ...
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2answers
58 views

If $G$ is a group with order $99$, it is cyclic by Sylow (isn't it?). I want to find a generator.

I have seen an argument in a specific case where $g,h\in G$ with $ord(g)=9$ and $ord(h)=11$ are used to create a generator through $f:=g^xh^y$ where $x, y \in \mathbb{Z}$ with $1=x9+y11$. Is this ...
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0answers
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Find multiplicative inverse and order of elements in group of units modulo $501$ and $4061$

Find the inverse of the following elements : Find $[91]^{-1}$, if possible (in $\Bbb Z^*_{501}$). Find $[3379]^{-1}$, if possible (in $\Bbb Z^*_{4061}$). Now, we have $\phi(501)=332$ and ...
2
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3answers
42 views

Inverse of elements in a group

If $x,y,z$ are elements of a group such that $xyz=1,$ then which of the following are true? $yzx=1$ $yxz=1$ $zxy=1$ $zyx=1$ I have found options 1 and 3 to be correct, but how to ...
1
vote
1answer
66 views

What groups are that? What does : mean?

What are the groups 2^6 : 3 . S_6 or 2^4 : A_8 ? Are they some subgroups of S_6 or A_8? I believe that 2 . A_n is the double cover of A_n, and "multiplying" with a number gives a covering group. But ...
0
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1answer
22 views

criteria for a short exact sequence of finite groups to be split

Suppose you have a short exact sequence of finite groups $1\rightarrow N\rightarrow F\rightarrow G\rightarrow 1$ such that $|G|$ and $|N|$ are coprime. Must the sequence be split? (Here I mean the ...
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2answers
46 views

If $a^n=e$ for exactly one $a$ then show that $a\in Z(G)$

The problem is, In a group $(G,\circ)$ if $a$ is the only element having order $n$ for some fixed $n\in\mathbb{N}$ then prove that $a\in Z(G)$. Remarks: Here $\mathbb{N}=\{1,2,\cdots\}$ ...
4
votes
5answers
79 views

$H,K$ are normal in $G$, then $HK$ is normal in $G$ (product of normal subgroups is normal)

This is a proof I couldn't find anywhere. Could somebody give me a help? I need this to show that $$\frac{H}{H\cap K}\cong \frac{HK}{K}$$ but to form the quotient group I need first to show that ...
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1answer
17 views

Definition of quasi-cyclic and full rational groups

In Unit Groups of Classical Rings by Karpilovsky, p.96, we can see this theorem: Let $G$ be a divisible abelian group. Then $G$ is a direct product quasi-cyclic and full rational groups. I want ...
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0answers
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List of simple roots in the H-basis for various Lie algebras?

There are four usual bases one can use to express the roots and weights of a given algebra. The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$. The ...
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3answers
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How do I proceed with this proof about order of elements in a group G?

Let a and b be elements of a group G. Let ord(a)=m and ord(b)=n. Prove if m and n are relatively prime, then no power of a can be equal to any power of b (except for e). My work so far: Suppose m ...
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1answer
31 views

Quotient group of product $HK$ over $H$ where $H,K$ are normal, is isomorphic to $K/({K\cap H})$

In order to prove that $H/({K\cap H}) \cong (HK)/K $ I found this question I understand that, in order to apply the first isomorphism theorem, I need to find a map from $H$ to $Kh$ such that $H\cap ...
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1answer
26 views

Why does the Principle of Well-Ordering imply a remainder of $0$ for the division algorithm?

I'm currently reading a text (Thomas W. Judson, Abstract Algebra - Theory and Applications) where the author proofs the theorem that every subgroup of a cyclic group is cyclic. The proof goes as ...
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2answers
41 views

Showing that the groups $S^1, SO_2, G, \mathbb R/\mathbb Z$ are isomorphic

I need to show that the groups $S^1, SO_2, G, H$ are isomorphic, where $S^1 = \{z\in \mathbb C\mid |z| = 1\}$ $SO_2 = \{A \in GL_2(\mathbb R)\mid AA^T = \text{Id,} \det(A) = 1\}$ $G ...
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3answers
41 views

Finding a normal and not normal subgroup of $S_3$

I'm being asked to find 2 subgroups of $S_3$, one of which is normal and one that isn't normal. I guess, to find the non normal subgroup is easier. I would do this by trial and error, but since the ...
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3answers
42 views

Galois group vs Permutation subgroups.[Confusion]

Okay my main problem rest with this quote from Rotmans group theory: Not every permutation of the roots of a polynomial $f(x)$ need correspond to some $\sigma\in Gal(K/F)$ but then he uses the ...
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4answers
83 views

Finding all groups of order $7$ up to isomorphism?

I'm learning group theory but I didn't learn any concepts of building groups. I know that there exists the identity group $\{e\}$, and the group with 2 elements: $\{e,a\}$. If I try to create a ...
4
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2answers
93 views

Let $a,b \in$ group $G$ such that $ab^3a^{-1}=b^2, b^{-1}a^2b=a^3$ Prove that $a=b=e$ (identity) [duplicate]

I got $ab^3=b^2a$ and $a^2b=ba^3$ by getting rid of the inverses via composing on both sides. I tried writing $a=aaa^{-1}$, which didn't help. Any suggestions please?
2
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1answer
53 views

Homomorphism of Free Groups

I am reading the theorem of homomorphism of free group from Fraleigh's text in $\S$36 and could only get a fuzzy idea at best: Let $G$ be generated by $A = \{a_i \mid i \in I \}$ and let $G'$ be ...
2
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0answers
23 views

Linear vs smooth actions of finite groups on spheres, Euclidean spaces and closed disks

I would like to know examples (with references, if possible) of the following: (1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ but admitting no effective linear action on ...
0
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1answer
23 views

Orbits for a subgroup $H$ of $G$ acting on $G$

Why the orbits for a subgroup of $H$ of $G$ acting on $G$ by left multiplication are the right cosets of $H$ in $G$?
2
votes
1answer
45 views

A Group Structure on given infinite set

If $X$ is an infinite set, can we always make it into a group in which every element has order $\leq 2$? Just for a countable set, I tried to define product with required property, but, because of ...
2
votes
1answer
38 views

Let $a,b \in$ group $G$ such that $ab=ba, \gcd(O(a),O(b))=1$. Prove that $O(ab)=O(a)O(b)$.

My attempt: Let $O(a)=m, O(b)=n$, then $mx+ny=1$ Let $O(ab)=p$, then using commutative property, $(ab)^p=a^pb^p=e$, which is the identity. Then $a^p=e, b^p=e$, hence, $m | p$ and $n | p$. So, ...