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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Commutator group of parabolic subgroups of $GL_n(q)$

Let $q$ be a prime power, $G=GL_n(q)$ and $P=q^{km}{:}(GL_k(q) \times GL_m(q))$ be a parabolic subgroup of $G$, where $k+m=n$. What is the commutator group $P'$ of $P$?
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478 views

Why is the collection of all groups considered a proper class rather than a set?

According to Wikipedia, The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and ...
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Abelian Groups of order 2000

Classify, up to isomorphism, all abelian groups of order 2,000, giving the standard form of each group in your list. (The standard form is also called the invariant factor decomposition.)
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710 views

Every group has a subgroup of prime order?

Is there a quick proof that given any finite group $ G $ with $ |G| = n$, it has a subgroup of prime order $ p \geq 2$? I've managed to prove the statement by writing down the unique prime ...
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293 views

For what algebraic curves do rational points form a group?

For what real algebraic curves do rational points form a group ? How does this relate to Jacobian Varieties ?
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Closure of Normal Subgroups of the Galois Group for an Infinite Galois Extension is Normal

Let K/F be an infinite Galois extension, and let N be a normal sub-group of Gal(K/F). Show that N closure is a normal subgroup of Gal(K/F).
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A reference writen in Russian

I'm reading a paper "Groups that can be represented as a product of two solvable subgroups" published in 1986 in Comm. Algebra. Since I do not understand Russian, I only read the abstract in this ...
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989 views

Show group of order $4n + 2$ has a subgroup of index 2.

Let $n$ be a positive integer. Show that any group of order $4n + 2$ has a subgroup of index 2. (Hint: Use left regular representation and Cauchy's Theorem to get an odd permutation.) I can easily ...
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31 views

computation in exact sequence of abelian groups

Suppose we have this exact sequence of abelian groups $$G_1\xrightarrow{f} G_2\xrightarrow{g}G_3\xrightarrow{h}G_4\xrightarrow{k}G_5.$$ I need to compute $G_3$ and I only know the expression of $f$ ...
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936 views

$a,b$ in $G$ has finite order. Then is the order of $ab, ba, a^{-1}b^{-1}$ with conditions finite?

Given that two elements $a,b$ in a group have finite order, are the following true: $ab = ba \implies ab$ has finite order. $ab$ has finite order $\implies ba$ has finite order. $ab$ has finite ...
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Which subgroups of $Gl_n(F)$ are normal? When is $Gl_n(F) \cong Sl_n(F) \times F^* I_n$?

Consider these subgroups of $Gl_n(F)$, where $F$ is a field and $n \epsilon \Bbb N$ $Sl_n(F)$ $Diag_n(F)$ $F^* I_n$ ($F^*$ times $I_n$, i.e. all $f \epsilon F^*$ times $I_n$) Questions: Show ...
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Sylow p-subgroups and normalizers

I am currently taking Abstract Algebra II and am stuck on the last question of my assignment. I asked the Professor about it and he said that he didn't remember how to do it and that there might be a ...
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456 views

Normal subgroups of the Special Linear Group

What is some normal subgroups of SL(2, R)? I tried to check SO(2, R), UT(2, R), linear algebraic group and some scalar and diagonal matrices, but still couldn't come up with any. So can anyone give ...
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417 views

Group of homomorphisms

Let $A$ be a finite, Abelian, additive group. Let $A^{*} = Hom(A, \mathbb{Q}/\mathbb{Z})$ denote the group of homomorphisms $f$ from $A$ to $\mathbb{Q}/\mathbb{Z}$. Take for granted that $A^{*}$ is an ...
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abelian group: $\mathrm{ord}(ab)|(\mathrm{ord}(a)\cdot\mathrm{ord}(b))$?

Let $H$ be an abelian group and $a,b\in H$ with $\mathrm{ord}(a)<\infty$ and $\mathrm{ord}(b)<\infty$. My question is why $\mathrm{ord}(ab)|(\mathrm{ord}(a)\cdot\mathrm{ord}(b))$ and why there ...
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88 views

Number of subgroups of order p

Let p be a prime number and consider $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}$. How many subgroups of order p does it have? Given any two subgroups $B_1, B_2 $ of ...
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136 views

Prüfer groups are countable

I have read that for any prime number $p$ the Prüfer $p$-group is countable. My question is: where can I find a proof of this fact? Thanks.
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Exponent of finitely generated group

Let $G$ be a finitely generated group with generators $X=\{g_1,g_2,\dots,g_m\}$ and $n$ be a positive integer such that the $n$-th power of every element in $X$ is the identity. Is it true that ...
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Attempt to prove that every subgroup is an equalizer

I'm trying to prove: For any subgroup $H$ of $G$, there is a group $T$ and homomorphisms $f,g:G\to T$ such that $f(x)=g(x)$ iff $x\in H$. My idea is to construct a group which contains two ...
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1answer
131 views

Prove that the order of an induced inner automorphism divides the element that induces it.

Let $a$ belong to a group $G$ and let $|a|$ be finite. Let $ø_a(x) = axa^{-1}$ for elements $x$ in $G$. Show that the order of $ø_a$ divides the order of $a$. Exhibit an element from a group for ...
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572 views

If $aH=bH$ implies $Ha=Hb$ then $aHa^{-1}=H$ for all $a\in G$.

The Question: If $aH=bH$ forces $Ha=Hb$ in a group G, show that $aHa^{-1}=H$ for all $a \in G$. My Attempt: I understand that $H$ must be a normal subgroup and that normal subgroups are closed ...
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Show that $x$ has larger order than $a$

This is exercise $2.35$ from Rotman's A First Course in Abstract Algebra. Let $G$ be a group and let a $a \in G$ have order $pk$ for some prime $p$, where $k \geq 1$. Prove that if there is $x \in ...
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Fix points of group action under base change

Let $A$ be a local noetherian domain. Let $M$ be a torsion free $A$-module equipped with an $A$-linear action of a group $G$. Let $\mathfrak{m}$ be the maximal ideal in $A$. Is the natural map ...
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Finding a new presentation for $H=\langle ab,abc\rangle$

Using Todd-Coxeter algorithm (a handy long calculations), I could find that the subgroup $H=\langle ab,abc\rangle$ of group $$G=\langle a,b,c|abcab^{-1}=bcabc^{-1}=cabca^{-1}=1\rangle$$ is of index ...
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$G$ is a nonabelian group with $[G : Z(G)] = n$. Show $\vert C_a \vert$ strictly less than $n$.

Prove: If $G$ is a nonabelian group with $[G : Z(G)] = n$, then every conjugacy class of $G$ has strictly fewer than $n$ elements. My approach so far: Observe that $Z(G) \leq C_G(a) = G_a \leq G$. ...
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79 views

Seeking a proof of: If any two Abelian groups of order $d$ are isomorphic, then $d$ is squarefree.

Suppose that $d \in \mathbb{N}$ satisfies the property that given any two Abelian groups $A_1, A_2$ of order $d$, $A_1 \cong A_2$. Prove that given any prime $p$, $p^2 \nmid d$. What can one say for ...
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110 views

Homomorphism $\mathbb{Z}^2 \to \mathbb{Z}^2$

I want to find a group homomorphism $f: \mathbb{Z}^2 \to \mathbb{Z}^2$ which satisfies $f(1,0) = (2,6)$. Can any such homomorphism be made into an isomorphism?
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Non-trivial homomorphism between multiplicative group of rationals and integers

Let $\mathbb{Q}^{\times}$ be the multiplicative group of non-zero rationals. Is there a non-trivial homomorphism $\mathbb{Q}^{\times} \to \mathbb{Z}$? In the same spirit, is there a homomorphism ...
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nilpotent group of class 2

Can you show that $\,exp(G/Z(G))=2 \Longrightarrow exp(G')=2$ ? My try is: clearly $\,G/Z(G)\,$ is abelian group,so G is nilpotent of class 2 and we have [G',G]=1 if x ϵ G' then [x,g]=1 ∀ g ϵ G so ...
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existence of hyperbolic groups

I perfectly understand that the Milnor-Schwarz lemma tells me that cocompact lattice in semisimple Lie groups of higher rank are not hyperbolic (in the sense of Gromov). But do there exist ...
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Please give me an example of a locally nilpotent group such that its derived subgroup is a p-group but its central factor is not torsion.

Please give me an example of a locally nilpotent group such that its derived subgroup is a p-group but its central factor is not torsion. Homework
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Complement of a subgroup

Let $G$ be a finite group. Suppose that every element of order $2$ of $G$ has a complement in $G$, then $G$ has no element of order $4$. Proof. Let $x$ be an element of $G$ of order $4$. By ...
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The various central extensions of $(G\times G)$ by $T$.

Let $G$ be a locally compact abelian group, isomorphic to $G^*$, its Pontryagin Dual. Let $T$ denote the unit circle in $\mathbb{C}$, where continuous morphisms $\chi: G\to T$ are the elements of ...
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Chernikov groups

A Chernikov group is a group $G$ that has a normal subgroup $N$ such that $G\over N$ is finite and $N$ is a direct product of finitely many quasicyclic groups. PlanetMath $(*)$ A periodic group of ...
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Explanation of first part of proof that if G is any group Z(G) is a normal subgroup of G.

Here's the proof of this fact from Schaum's book on group theory - $1 \in Z(G)$ since $1g = g1$ for all $g \in G$. Consequently $Z(G) \neq \emptyset$. If $g_1,g_2 \in Z(G)$ and $g \in G$, then ...
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What's the difference between the center of a group and a normal subgroup?

It seems the definition of the center of a group and a normal subgroup are the same so I'm wondering what the difference is between the two? A group $H$ is normal in $G$ iff $Hg=gH$ for all $g \in ...
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Where can I find the original papers by Frobenius concerning solutions to $x^n = 1$ in a finite group?

A theorem proven by Frobenius states that If $n$ divides the order of a finite group $G$, then the number of solutions to $x^n = 1$ in $G$ is a multiple of $n$. Articles discussing this theorem ...
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Is every group homomorphism of the rationals an isomorphism

Is it true that every non trivial group homomorphism from $\mathbb Q$ to $\mathbb Q$ is a group isomorphism. The trivial homomorphism being the map that sends every rational to $0$.
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Would you please construct a locally nilpotent group which is not nilpotent?

Would you please construct a locally nilpotent group which is not nilpotent? An example of a locally nilpotent group which is not nilpotent?
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1answer
82 views

Arc transitivity of the complete graph

Recall that a graph $G$ is arc transitive if the natural action of $\mathrm{Aut}(G)$ on $A(G) = \{ (u,v) | \{u,v\} \in E(G)\}$ is transitive. In other words, given $(u,v),(u'.v') \in A(G)$ one finds ...
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How to show that these groups are not isomorphic?

Let $G$ is an abelian group and $tG$ is its torsion subgroup. If $p$ is a prime, how to show that: $$t\bigg(\prod_{n=1}^{\infty}\mathbb Z_{p^n}\bigg)\ncong\sum_{n=1}^{\infty}\mathbb Z_{p^n}$$ I ...
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First Isomorphism Theorem

Theorem: Let $G$ and $G'$ be groups and let $f:G\to G'$ be a group homomorphism. Then $G/\textrm{ker}\, f \cong\textrm{im}\, f$. My question is how to understand this theorem intuitively.
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Adapting a proof on elements of order 2: from finite groups to infinite groups

Consider the following problem, appropriate for a first course in Group Theory: Problem: Prove that there cannot be a group with exactly two elements of order $2$. General Proof: Suppose for the ...
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A bound for the exponent of the Schur multiplier of group G

Let $G$ be a finite group and $M(G)$ be the Schur multiplier of $G$. Show that the exponent $\exp(G)$ of $G$ divides the product of the exponents $\exp(M(S_p))$ of the Sylow $p$-subgroups $S_p$ of ...
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If every proper quotient is finite, then $G\cong\mathbb Z$

Here is my problem: Let $G$ is an infinite abelian group. Prove that if every proper quotient is finite, then $G\cong\mathbb Z$. And here is my incompleted approach: I know that the quotient ...
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Conjugacy classes of SO3 and O3

I'm trying to find the conjugacy classes of SO3 and O3. How do I do this? SO3 consists of all rotations around any axis in three dimensions but how do I determine which are conjugate?
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Does S5 have subgroups isomorphic to D5 and D6? [duplicate]

Possible Duplicate: Questions on Symmetric group of degree 5 I was told that $S_5$ has subgroups isomorphic to $D_5$ and $D_6$, however when I looked it up here I didn't see any mention of ...
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Show that if V is normal in G then UV is a subgroup of G

Suppose $U$ and $V$ are subgroups of a group $G$. Show that if $V$ is normal in $G$, then $UV = \{\,uv\,|\,u \in U, v \in V\,\}$ is a subgroup of $G$. I have shown the identity axiom. For the ...
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How can we make $\mathbb{R}^n$ into a multiplicative group?

Are there any sorts of multiplicative group structures we can put on set $\mathbb{R}^n$? By multiplicative, I mean that it should be compatible with the natural scalar multiplication on ...
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Showing that a homomorphism between two groups occurs

The problem is: Let G and G' be groups, and let H and H' be normal subgroups of G and G', respectively. Let $\phi$ be a homomorphism of G into G'. Show that $\phi$ induces a natural homomorphism ...