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 $G$ 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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1answer
43 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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1answer
42 views

Prove that $I=\{x \in R : ax=0\}$ is a subring of $R$

To do so we just need to prove that I is a ring itself. By requirements needed to be a ring, $(I,+)$ is an abelian group: Let $x,y \in R$ then we have that $ax=0=ay$. So $ax+ay=0=ay+ax$. $\ast$ is ...
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2answers
94 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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1answer
76 views

finite simple groups and free groups

It is well known that any group is a homomorphic image of a free group. I want to know more about this theorem when $G$ is a finite simple group. Does there exist any reference to state about it? ...
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0answers
47 views

Calculate the conjugancy classes for $D_4$

Calculate the conjugancy classes for $D_4$ $$D_4=\{e,r,r^2,r^3,s,sr,sr^2,sr^3\}$$ The conjugancy class of $e$ is just $\{e\}$. To find the conjugancy class of $r^n$, note that when we conjugate $r^n$ ...
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0answers
63 views

How to construct explicit matrix representations of $\mathfrak{su}(3)$

I'd like to implement an algorithm which produces matrix representations of the (complexified) Lie Algebra $\mathfrak{su}(3)$ on carrier spaces with arbitrary highest weight vector; i.e. 8 $n\times n$ ...
3
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0answers
89 views

Automorphism(Galois groups) and galois theory

I've been stuck on two last parts for two different questions, can someone please help me with these. The first question is: Let $\sigma\in Aut(L/\mathbb{Q})$, where $L$ is some subfield of ...
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1answer
79 views

Let G be a group of order 2n with n being an odd integer. Prove that G has a normal subgroup of order n.

I have been trying to work this problem out for a bit and am stuck. Does anybody have any ideas how to proceed or solve this? I think this has something to do with rings and fields but I can't seem ...
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3answers
46 views

Prove that $C_{A_7}((567))\cong H \times A_4$

Let $H=(<567>)\subset{A_7}$. And let $C=C_{A_7}((567))$ denote the centralizer of $(567)$ in $A_7$. Prove that $C_{A_7}((567))\cong H \times A_4$ I'm fairly certain that I can use the First ...
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4answers
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False homework proof?: The image of an element has the same order.

I have this assignment from a homework that I'm pretty sure is wrong. It asks me to prove Given a group homomorphism $\phi: G\rightarrow G'$, if $g\in G$ has order $k$ then so does $\phi(g)$. I ...
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0answers
20 views

On a n-Engel Verbal Subgroup be perfect

Given a Group $G$, the $n$-th Engel Word defined by $[x,\ _0{x}]=x: \ [x,_{ \ n}y]=[[x,_{\ {n-1}}y],y]$ in the group $G$ consists by substituting group elements for the determinates. The Group ...
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1answer
25 views

subgroups with cycles

Let Sym(n) denote the symmetric group on n letters and let H be a subgroup of Sym(n) . Suppose that H contains a k cycle for each value of k from 2 through n . This should be enough to ...
2
votes
2answers
65 views

Show that a subgroup $K$ is normal [duplicate]

Let $K$ be the subgroup $K=\{e,(1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\}$ in $S_4$. show that $K$ is normal in $S_4$. show that $S_4/K \cong S_3$. I know that I could try to prove this directly ...
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1answer
61 views

all Sylow subgroups of $GL_n(\mathbb{F}_q)$

Can you give some references to find all Sylow subgroups of $GL_n(\mathbb{F}_q)$? I know that upper triangular matrices with diagonal's 1 is a Sylow $p$-subgroup where $q=p^n$. But how about the other ...
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0answers
57 views

Help with proof for problem 14 chapter 5 Dummit and Foote.

For any group $G$ define the dual group of $G$ (denoted $\hat{G}$) to be the set of all homomorphisms from $G$ into the multiplicitive group of roots of unity in $\mathbb{C}$. Define a group operation ...
2
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1answer
54 views

Polygons with fixed number of axes of symmetry

Let $k$ be a positive integer. Suppose a polygon has exactly $k$ axes of symmetry. How many sides may the polygon have? A regular $n$-gon has $n$ axes of symmetry, so one answer is $k$. What are ...
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1answer
46 views

Prove that $Q_8 \cong \langle a, b \mid a^4, a^2b^{-2}, aba^{-1}b \rangle$

Prove that if $G = \langle a, b \mid a^4, a^2b^{-2}, aba^{-1}b \rangle$, then $G \cong Q_8$. I started by trying to define a homomorphism $\varphi: F(a,b) \to Q_8$ by $\varphi(a) = i$, $\varphi(b) = ...
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0answers
58 views

Order of conjugate map

Let $\sigma :L\rightarrow L$, s.t $\sigma (\alpha)=\bar{\alpha}$. I've been asked to show that $\sigma\in Aut(L/K)$(the set of all automorphisms for the field extension) has order 1 or 2. I'm ...
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1answer
86 views

Show that every subgroup of the quaternion group Q is a normal subgroup of Q [closed]

How should I start to show this. Can anyone give the specific steps?
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2answers
108 views

How many elements of S8 have the same cycle structure as (12)(345)(678)?

Should I do this by counting all the numbers using a factorial and then multiply by 8? Can any one show me the specific steps?
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2answers
53 views

Definition of dual group.

The following definition is given in Dummit and Foote's Abstract Algebra on page 146. For any group $G$ define the dual group of $G$ (denoted $\hat{G}$) to be the set of all homomorphisms from $G$ ...
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1answer
142 views

Show that a group of order $180$ is not simple.

What i deduced is that $n_5=1,6$ or $36$. We are done if $n_5=1$. If $n_5=36$ we $N_G(P)=P$ for any Sylow $5$-subgroup P as $|N_G(P)|=\frac{180}{36}=5$ and $P$ is abelian cyclic so by Burnside ...
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1answer
30 views

Implications of group action deffinition

As given definition of group action goes like this : For any group G and X is a set then $\phi : G \times X \to X$ and such that $\phi(1,x)=x$ and $\phi(g,\phi(h,x))=\phi(gh,x) $ for every $x \in X$ ...
5
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1answer
114 views

Is there an interpretation of higher cohomology groups in terms of group extensions?

1) Consider a group $G$ and a $G$-module $A$. Then it is well-known that there is a $1-1$ correspondence between elements of $H^2(G,A),$ and group extensions $1\rightarrow A \rightarrow H\rightarrow ...
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votes
1answer
29 views

Suppose $x$ and $y$ are in the same orbit. Show that $G_x$ and $G_y$ are conjugate subgroups.

Let $X$ be a $G$-set and $x,y \in X$. Let $G_x$, $G_y$ be the stabilizers of $x$ and $y$ respectively. Suppose $x$ and $y$ are in the same orbit. Show that $G_x$ and $G_y$ are conjugate subgroups. ...
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1answer
25 views

CHECK: Show that $N_G(aHa^{-1})=aN_G(H)a^{-1}$ for $a \in G$.

Show that $N_G(aHa^{-1})=aN_G(H)a^{-1}$ for $a \in G$. Hint: When working this problem, I found that in showing the two sets equal I had to be extremely careful. Try not to make big jumps. At one ...
3
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1answer
40 views

Find $n$ such that $(\mathbb{Z}/n\mathbb{Z})^\times \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$.

In the ring $(\mathbb{Z}_n, +, \times)$, we know that only $\phi(n)$ of the elements have multiplicative inverses which form the multiplicative group $\mathbb{Z}_n^\times$. However, since $\phi(n)$ ...
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votes
1answer
151 views

Isomorphism Type

Given $G=\mathbb{Z}_{16}^{\times}$ then how does one go about to determine the isomorphism type of $G/ \langle15 \rangle$ and $G/\langle 9 \rangle$? Question: When you say $G/⟨15⟩$ has 4 ...
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2answers
80 views

Isomorphisms of two subgroups in $S_6$

Is there any nice group action to see why groups $S_2 \times S_4$ and $S_2 \wr S_3$ of order 48 are isomorphic? Or is this "just" an abstract property which becomes invisible when we switch to ...
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0answers
58 views

Generalization of a property of finite nilpotent groups

Let $G$ be a finite nilpotent group and $M$ be a maximal subgroup of $G$. If $H$ is a proper non-trivial subgroup of $G$ such that $H\not\leq M$, then we can show that $H\cap M$ is a maximal subgroup ...
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1answer
43 views

Conditions for finiteness of group in geometric group theory

Are there any sufficient conditions in geometric group theory for a group to be finite? Are there any necessary conditions?
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43 views

Structure of an affine group as a semidirect product

Let $A(n,K)$ stand for the group of affine transformations on the affine space $K^n$, where $K$ is a field. As an outer semidirect product, one has $A(n, K) = K^n \rtimes \mathit{GL}_n(K)$, where the ...
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1answer
49 views

Explaining That there is no nontrivial ring homomorphism between Z and nZ

My instructor wrote in his notes the following example: "As groups (Z,+) and (nZ,+) are isomorphic. As rings is there any nontrivial homomorphism $\phi$: Z->nZ? The answer is no and he gives the ...
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1answer
114 views

Union of all finite cyclic groups

Let $C_n$ be the cyclic group of order $n$. I'm trying to investigate properties of $\bigcup_{n=1}^{\infty} C_n$ It seems obvious that this is a group, but I don't really know much else about it. ...
3
votes
2answers
61 views

For $\phi$ as a homomorphism not onto show not normal subgroup [duplicate]

I am just a little unclear about what I am supposed to do here. Let $\phi: G $ to $J$ be a homomorphism onto all of $J$. Let $H$ be a normal subgroup of $G$ and let $K=\phi(H)$ be the image of $H$ ...
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0answers
45 views

Galois Group of a rational polynomial

How could I do to calculate the Galois group of a polynomial with rational coefficients ? Have you some links or books where I could find an answer ?
2
votes
1answer
59 views

Find a non-principal ideal (if there exists any) in the rings Z[x], Q[x], Q[x, y]

I know that $Q$ is a field, which makes $Q[x]$ a PID, which means there are none. I'm having trouble with the notation for ideal generators, and i know the $Z[x]$ has to do with something that looks ...
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1answer
36 views

Question on normal subgroups

$H=\{e, (12)(34)\}, K=\{e, (12)(34), (13)(24), (14)(23)\}$. Show the following: a) $H$ is normal in $K$. b) $K$ is normal in $S_{4}$. c) $H$ is not normal in $S_{4}$.
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Irreducible representations of inhomogeneous linear transformations over $\mathbb{F}_q$

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $G$ be the group of inhomogeneous linear transformations over $\mathbb{F}_q$, that is $x\mapsto ax+b$ for some $a\in \mathbb{F}_q^*$ and ...
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1answer
36 views

Show that a polynomial f(x) over a field k is irreducible if and only if the polynomial f(x + 1) is irreducible.

I am very much unsure what definitions and formulas are relevant for this question. I've toyed around with the lemma "An element a ∈ R is a root of a polynomial f ∈ R[x] if and only if (x − a) divides ...
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3answers
58 views

Why is this map well-defined?

Let $G$ a finite group and $H$ and $K$ two sub-groups of $G$. Why is the map \begin{array}{rcl} \Psi: G/({H\cap K}) & \longrightarrow &G/H \times G/K \\ g(H\cap K) & \longmapsto ...
3
votes
1answer
91 views

Representation of the symmetry group (rotations) of the icosahedron

Suppose $I$ is the set of vertices of the regular icosahedron, here is a link of the icosahedron: http://www.werheit.mynetcologne.de/icosaeder.gif Let $F(I)$ be the space of complex functions on $I$, ...
2
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0answers
32 views

Please help to understand a proof that any Galilean transformation can be written as composition of three (standard) transformations

I'm trying to understand one line I saw in a proof that any element of the Galilean group (i.e. any Galilean transformation) on the canonical ($\mathbb R \times \mathbb R ^3$) Galilean space can be ...
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5answers
58 views

If H a subgroup, and the left and right cosets are equal, does this mean that H is the center?

Theorem: If $H$ is a normal subgroup of $G$, then $aH=Ha$ for every $a \in G$. If H a subgroup, and the left and right cosets are equal, does this mean that H is the center? Isn't the definition ...
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1answer
34 views

Prove that for each $\sigma \in Aut(S_n)$ $n \neq 6$. $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), …, \sigma(1,n)=(a,b_n)$

I am reposting it getting insufficient help from the previous post (Although I got some hint) Prove that for each $\sigma \in Aut(S_n)$ $n \neq 6$. $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), ...., ...
2
votes
1answer
108 views

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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votes
0answers
75 views

Finding 1st and 2nd homology groups using augmentation ideal

Let $G$ be a group and $I_{G} = ker(\epsilon)$ be the augmentation ideal where $\epsilon: Z[G] \rightarrow Z$ is the augmentation ring homomorphism $<g> \rightarrow 1$. Let $f: I_{G} ...
2
votes
3answers
84 views

When does a ring map $R\to S$ produce a group epimorphism $GL_n(R)\to GL_n(S)$?

Let $R$ and $S$ be rings with $1$ (not necessarily commutative) and $f:R\to S$ a ring homomorphism preserving $1$. Let $\bar{f}$ be the ring map $M_n(R)\to M_n(S)$ given by $f$ acting on the matrix ...
3
votes
1answer
37 views

Can kernel of homomorphism tell you when a group action cannot be constructed?

I understand that for every action of group $G$ on a set $X$, there is a homomorphism: $$G\rightarrow S_X$$ It seems to me that this can be used to rule out many possible actions. For example, a group ...