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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Are all normal subgroups Abelian?

If $H \subset G$ is a normal subgroup of G, => $xHx^{-1} = H$ or $xH = Hx$ for all $x \epsilon G$ => $xH = Hx$ for all $x \epsilon H$ Hence, all normal subgroups of a group are themselves Abelian? ...
6
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2answers
88 views
+100

Describe the orbits of the action.

So $L$ denotes the set of oriented straight lines through the origin in $\mathbb{R}^{2}$ (that is, straight lines with a preferred direction, indicated by an arrow). The group $$(\mathbb{R},+)\cong ...
6
votes
1answer
43 views

Group $G$ acts faithfully on a set $X$ of 5 objects? [duplicate]

A group $G$ acts faithfully on a set $X$ of 5 objects. The action has two orbits: one of size 2, and one of size 3. What are the possibilities for the group $G$? I believe the right step is ...
1
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0answers
57 views

How to show a group is infinite group by constructing epimorphism?

Consider a group G with representation $$\langle a,b|abab^{-1}a^{-1}b^{-1}\rangle$$ Prove that this group is an infinite group There is similar question here (Finding the kernel of an epimorphism onto ...
3
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0answers
50 views

What do you call the category of groups with surjections (injections)?

Take the category $Grp$ and throw out all morphisms that are not surjections. I think what's left is still a category. The composition of two surjections is surjective, and identities exist ...
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1answer
23 views

Proving a group homomorphism between two groups with different operations

Here is a rather simple algebra question: Suppose $(G,\star)$ and $(H,\circ)$ are groups and $\varphi$ is a map from $G\rightarrow H$. Am I correct in understanding that to show $\varphi$ is a ...
0
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0answers
10 views

The semidirect product of two hypercentral groups [on hold]

I have a question: Let $G$ be a group such that G=$H\rtimes K$ and both H and K are hypercentral subgroups of $G$. Does $G$ also hypercentral group? We recall that a group G is hypercentral if ...
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1answer
20 views

$\mathrm{GL}(V)$ acting on bilinear maps

Let $V,W$ be a finite dimensional real linear spaces, and $\beta:V\times V\to W$ a bilinear map. Then my notes claim that for $g\in\mathrm{GL}(V)$ we define $g\cdot\beta(u,v)=\beta(g^{-1}u,g^{-1}v)$, ...
-1
votes
0answers
13 views

Normalizer of invertible upper triangular matrices [on hold]

I have stumbled upon the following question: I am given the group $G=GL_n(\mathbb{C})$. If $B\subset G$ is the subgroup of all invertible upper triangular matrices, then $N(B)=\{g\in ...
0
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0answers
22 views

Problems in understanding the result of this composition of automorphisms

Let be p a prime number and let be $|x|=p^{n-1}$. I consider $\alpha \in {\rm Aut}(\langle x \rangle)$ defined in this way: $\alpha : x \mapsto x^{1+p}$. Let be $\gamma=\alpha^{p^{n-3}}$. I don't ...
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2answers
35 views

Free cocompact action of discrete group gives a covering map

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...
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1answer
17 views

For any permutation $ \sigma \in S_n$, $(σ(1) − 1)(σ(2) − 2) . . . (σ(n) − n)$ is even when $n$ is odd [on hold]

Let σ be a permutation of ${1, 2, 3, . . . , n}$, n odd. I want to show that $(σ(1) − 1)(σ(2) − 2) . . . (σ(n) − n)$ is even. Thank you.
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1answer
22 views

Finding a finite $p$-group of nilpotency class $n$ for each $n>1$

There's a problem in Rotman's group theory book that goes For $n\ge1$, let $G_n$ be a finite $p$-group of class $n$. Define $H$ to be the group of sequences $(g_1,g_2,\dots)$ for $g_n\in G_n$ and ...
2
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1answer
56 views

Prove that if $x$ is conjugate to $x^{-1}$ and $y$, that $y$ is conjugate to $y^{-1}$

The full question: Prove the following: a) If $x$ is conjugate to $x^{-1}$ and $y$, then $y$ is conjugate to $y^{-1}$ b) If $x$ is conjugate to $x^{-1}$ in a finite group, $G$, and $x \neq x^{-1}$, ...
2
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2answers
44 views

Find all the homomorphisms from $D_8 \to \mathbb{C}^\times$

Find all of the homomorphisms from $D_8$ to $\mathbb{C}^\times$. So far I have: $\phi : D_8 \rightarrow \mathbb{C}^\times$ $\phi(a)^4 = 1$ so $\phi(a) = \pm 1, \pm i$ $\phi(b)^2$ = 1 so ...
1
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1answer
47 views

$Aut(G)$ is abelian if and only if $G$ is cyclic. [duplicate]

Problem says: Let G be an abelian group. Prove that Aut(G) is abelian if and only if G is cyclic. And I solved $(\Leftarrow)$ direction as follow: Suppose that $G$ is cyclic. Then ...
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0answers
25 views

Pronormality of Sylow subgroups

I need help on proving that every Sylow subgroup of a finite group is pronormal. A subgroup $H$ of a group $G$ is said to be pronormal if for each $g\in G$, the subgroups $H$ and $gHg^{-1}$ are ...
0
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0answers
10 views

Gell-Mann Matrices with dot and cross product

We know that, given $\vec{a}$ and $\vec{b}$, then $(\vec{a}\cdot\vec{\sigma})(\vec{b}\cdot\vec{\sigma})=(\vec{a}\cdot\vec{b})\mathbb{I}+i(\vec{a}\times\vec{b})\cdot\vec{\sigma}$. $\sigma_i$ are the ...
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0answers
15 views

Inverse of an element in an external direct product

Let $G = \mathbb{Z}_{4}\times S_{5}$ What is the inverse of $\left ( 3,\left ( 1,2 \right )\left ( 3,5 \right ) \right )?$ The inverse of any elements a in $\mathbb{Z}_{4}$ is 4-a. So the inverse of ...
0
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1answer
23 views

Order $4$ subgroup of alternating group $A_4$

I ran into the following problem: Let $H$ be the subgroup $H = \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}$ in $G = A_4 = H \cup \{(1\, 2\, 3), (1\, 3\, 2), (1\, 2\, 4), (1\,4\,2), ...
2
votes
1answer
65 views

Is $\operatorname{Mat}(2,\mathbb{R})$ isomorphic to $\operatorname{Mat}(3,\mathbb{R})$? [on hold]

As the title. Is $\operatorname{Mat}(2,\mathbb{R})$ isomorphic to $\operatorname{Mat}(3,\mathbb{R})$? I cannot seem to find any bijection that will do. Thanks. $\operatorname{Mat}(n,\mathbb{R})$ ...
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1answer
28 views

Quotient ring $(\mathbb{Z}_4 \times \mathbb{Z}_6)/S$

Consider the ring Z4xZ6 with +6, *6, and +4, *4 in appropriate coordinates and S={(0,0),(2,0),(0,3),(2,3)}. Would the elements of the quotient ring Z4 x Z6 / S be: S+0 (trivial set above), ...
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0answers
17 views

About finitely generated torsion-free nilpotent groups

The only finitely generated torsion-free nilpotent group I know is the Heisenberg group: $$H= \left\{ \left(\begin{array}{ccc} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1 ...
-3
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0answers
32 views

Let G be a group, and suppose that H is a subgroup of G of order 5, and no other subgroup of G has order 5. Show that H is a normal subgroup of G. [on hold]

Let G be a group, and suppose that H is a subgroup of G of order 5, and no other subgroup of G has order 5. Show that H is a normal subgroup of G. Picture of my proof I got 5/10 on this problem. It ...
-3
votes
0answers
67 views

Using Gap system. [on hold]

I'm new at the GAP. Probably I can't use this system, what I type doesn't work. For instance why the following doesn't work? for i in [1..1160] do Print("Processing semigroup number ",i,"\n"); ...
0
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1answer
27 views

Number of conjugates of x in K a subgroup of G.

Let $G$ be a group, let $K$ be a subgroup of $G$ and $x$ $\epsilon$ $G$. Use the technique that you know to show that the number of distinct conjugates of $x$ by elements of $K$ is $[K : K \bigcap C_G ...
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0answers
23 views

Show that ${C_\infty }/\left\langle {{c^n}} \right\rangle \simeq {C_n}$

Let ${C_\infty } = \left\langle c \right\rangle $ be an infinite cyclic group. Show that if $n > 0$, then $${C_\infty }/\left\langle {{c^n}} \right\rangle \simeq {C_n}$$where ${C_n}$ is a ...
-2
votes
1answer
50 views

Subgroups of $\mathbb{Q}/ \mathbb{Z}$ [on hold]

I was asked to show in my exam that all the subgroups of $\mathbb{Q}/ \mathbb{Z}$ are of the form $\{\mathbb Z + a/p^i\}$ where $p$ is a prime number and $0\leq a \lt p^i$, where $i$ varies over all ...
3
votes
3answers
171 views

Show that $(\mathbb Z[x],+)$ and $(\mathbb Q_{>0},\cdot)$ are isomorphic groups [closed]

Let $(\mathbb{Z}[x],+)$ be the additive group of polynomials with integer coefficients and $ (\mathbb{Q}_{>0},\cdot)$ the multiplicative group of positive rationals. Show these groups are ...
1
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1answer
21 views

Isomorphism between the group $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$ [duplicate]

In one of my assignment, I was told that $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$ are isomorphic, with $$\phi (\sum_{k=0}^n a_k x^k) = \prod_{k=0}^n p_k^{a_k}$$ is a one-to-one surjective ...
0
votes
1answer
21 views

Polynomial roots conditions vary with coefficients

Polynomial equation $\sum_{i=0}^4 p_i x^i=0$ have the following root conditions: 1) $a_0 \pm b_0 i, a_1 \pm b_1i$ 2) $a_0 \pm b_0 i, a_1, a_2$ 3) $a_0, a_1, a_2 \pm b_2i$ 4) $a_0, a_1, a_2, a_3$ I'm ...
0
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4answers
60 views

Why is the 'Law of Cancellation' for groups only an implication?

It is easy to see that for a Group $G$ and $a,b \in G$ $ab = ac \Rightarrow b = c$ (See also here) But what is about the other direction? That is: $b = c \Rightarrow ab = ac$ Does this ...
0
votes
1answer
27 views

Constructing representation of $G$

Say we are given an arbitrary group $G$ and an arbitrary vector space $V$ over some field. How can we construct a representation of $G$ on some vector space from this data? Initially I wanted to ...
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0answers
15 views

The nilpotency class of the dihedral group $D_{2^n}$ is $n-1$

I want to prove that the nilpotency class of the dihedral group $D_{2^n}$ of order $2^n$ is $n-1$ for $n>2$. I've tried to prove this by induction. I have proven that the nil potency class of ...
0
votes
0answers
35 views

Is the following function a homomorphism?

Is the following function a group homomorphism? $f:G\to G'$, where $G=(\Bbb{R},*)$ and $G'=(\Bbb{R}^+,o)$, and $f(x)=e^x$.
-3
votes
0answers
33 views
1
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1answer
57 views

Non-abelian group which squares to equal the identity element?

Does there exist a non-abelian group $G=\{e,g_1,g_2,...,g_n\}$ with order $n+1$ s.t. \begin{align} (g_1 \dots g_n)^2 = e \end{align} Also, does this change if we say that every element in $G$ is its ...
0
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3answers
1k views

limit of square root of function and example of group

can somebody help in getting the value of $\displaystyle\lim_{x\to 9}\frac{\sqrt{f(x)}- 3}{\sqrt{x}-3}$, if $f(9) = 9$ and $f(9) = 4$. Though the problem seems to be very simple, when I tried the ...
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1answer
63 views

What is $gnu(18,480)\ $?

Probably, the number of groups of order $18,480$ can only be determined with GAP. But I may be wrong, so the question should not be understood to be purely computational. A better lower bound than my ...
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1answer
26 views

Is it right to say: A finite semigroup always contains an idempotent element? [closed]

Is there a theorem which says: If an operation on a set is associative then the set contains idempotent?
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0answers
12 views

Schur-Zassenhaus theorem for abelian normal Hall $\pi$-subgroup of $G$ [closed]

I am trying to prove part (b) of the Schur-Zassenhaus theorem which states that if $H$ or $G/H$ is solvable, any two complements of $H$ in $G$ are conjugate in $G$. I am unable to find a proof for ...
0
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1answer
25 views

How Many Orbits Does The Symmetric Group Action Sym(6) On Itself Have? [closed]

Let the group action $g \hookrightarrow X$ Let's define the following terms: orbit: Let $x \in X$ it's orbit is the set $G.x =${$g.x | g \in G$}$\in X$ Sym(X): It's the set of bijections $X ...
2
votes
1answer
28 views

Calculating the number of elements of a given order in a group of permutations.

Let $S$ denote the group of all those permutations of the English alphabet that fix the letters T, E, N, D, U, L, K, A, and R. Other letters may or may not be fixed. Show that $S$ has elements ...
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votes
1answer
55 views

Show that ker(f) = M. [closed]

Let $\mathbb{Z} \times \mathbb{Q}$ be the group of ordered pairs $(x, y)$ with $x \in \mathbb{Z}, y \in \mathbb{Q}$ under component-wise addition. Fix $m \in \mathbb{Q}$ and let $M \subset \mathbb{Z} ...
1
vote
1answer
22 views

Geometrical interpretation of the conjugacy of triangle groups.

Let $\triangle$ and $\triangle'$ be two hyperbolic triangles of respective angles $\alpha,\beta,\gamma$ and $\alpha',\beta',\gamma'$. Let us suppose that the triangle subgroups ...
0
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0answers
23 views

isomorphic permutation groups with same cycle index [closed]

Is there two nonidentical isomorphic permutation groups with same cycle index? (Two permutation groups A and B on sets X and Y, respectively, are said to be identical, if there is a function 1-1 map ...
2
votes
1answer
31 views

Regarding those $\mathbb{Z}$-modules whose every finite subset generates a finite submodule.

Let $X$ denote a $\mathbb{Z}$-module (aka an abelian group). Then $X$ may or may not satisfy: $(*)$ for all finite sets $F \subseteq X$, the module generated by $F$ is finite. This properly ...
1
vote
1answer
42 views

If the order of $G$ is odd, show that for any $y \in G$ there is a unique $x \in G$ such that $x^2=y$

This question seems to be a typical one, but I am not sure about the uniqueness part. My attempt Suppose $|G|$ is odd. By Lagrange's Theorem, we have all elements in $G$ is of odd order. In ...
2
votes
1answer
25 views

General property of Fitting series

Let $G$ be a finite solvable group. $F(G)$ (Fitting subgroup) is defined to largest normal nilpotent group contained in $G$. Then $F_2(G)$ is defined to be inverse image of $F(G/F(G))$. i.e ...
1
vote
1answer
33 views

Prove that $a . g = g^{-1}.a$ and $g . a = a . g^{-1}$ hold for any group and any action defined.

Dummit and Foote page 129 claims that for arbitrary group actions that if we are given a left group action of $G$ on $A$ then the map $A \times G \to A$ defined by $a . g = g^{-1}.a$ a is a right ...