Questions tagged [group-theory]
For questions about the study of algebraic structures consisting of a set of elements together with a well-defined binary operation that satisfies three conditions: associativity, identity and invertibility.
6,983
questions
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Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$.
Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$.
I can find all the infinite subgroups of the form $n\mathbb{Z}\times m\mathbb{Z}$, where $n,m$ run over $\mathbb{Z}$. But I don't know how to write ...
19
votes
3
answers
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Counterexample: $G \times K \cong H \times K \implies G \cong H$
I am having a hard time finding a counterexample for the statement:
$G \times K \cong H \times K \implies G \cong H$
I think this should be true for abelian, finite groups. But is this true in ...
17
votes
3
answers
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Is the dihedral group $D_n$ nilpotent? solvable?
Is the dihedral group $D_n$ nilpotent? solvable?
I'm trying to solve this problem but I've been trying to apply a couple of theorems but have been unsuccessful so far. Can anyone help me?
15
votes
3
answers
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Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$?
I know that given a group $G$ and two normal subgroups $H,K \subset G$ then it is not true that:
"if $H \cong K$ then $ \frac{G}{H} \cong \frac{G}{K} $ (the counterexample is quite easy with products ...
14
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4
answers
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free groups: $F_X\cong F_Y\Rightarrow|X|=|Y|$
I'm reading Grillet's Abstract Algebra. Let $F_X$ denote the free group on the set $X$. I noticed on wiki the claim $$F_X\cong\!\!F_Y\Leftrightarrow|X|=|Y|.$$ How can I prove the right implication (...
13
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3
answers
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Classification of groups of order 30 [duplicate]
How do I find all the groups of order 30? That is I need to find all the groups with cardinality 30. I know Sylow theorems.
13
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3
answers
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Number of Homomorphisms from $\Bbb{Z}_m$ to $\Bbb{Z}_n$
This question coutesy of Allan Clark's "Elements of Abstract Algebra" (60$\zeta$).
Find the number of homomorphisms from $\Bbb{Z}_m\to \Bbb{Z}_n$ as a function of $m$ and $n$.
This is stumping me, ...
12
votes
2
answers
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Order of products of elements in a finite Abelian group
We want to show that if $a,b\in G$ where $G$ is a finite Abelian group, we have
$\operatorname{LCM}(|a|,|b|) = |ab|$ given that $ab \neq e$.
How I approached this question was by saying let $\...
9
votes
3
answers
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Finite Index of Subgroup of Subgroup
Prove the following: If $H$ is a subgroup of finite index in a group $G$, and $K$ is a subgroup of $G$ containing $H$, then $K$ is of finite index in $G$ and $[G:H] = [G:K][K:H]$.
So this is ...
7
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4
answers
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Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
I am completely stuck at the question
Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order 2 in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $n = |xy|.$
I have ...
6
votes
1
answer
547
views
Subextension of a field with Galois series of subextensions of prime degree
Let $p$ be a prime number, and $E/F$ be a field extension. Suppose $E/F$ has a finite series of subfields
$$
F = E_0 < E_1 < \cdots < E_n = E
$$
with $E_i / E_{i-1}$ Galois of degree $p$ for ...
3
votes
5
answers
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Isomorphism $f$ preserves the order of an element?
Let's say $f$ is an isomorphism $f:G \rightarrow G'$, where $G$ and $G'$ are multiplicative groups.
Then for $x\in G$, if $f(x) = x' \in G'$. Do we have always have $\text{ord}(x) = \text{ord}(x')$? ...
68
votes
4
answers
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Center-commutator duality
I'm reading this article by Keith Conrad, on subgroup series. I'm having trouble with a statement he does at page 6:
Any subgroup of $G$ which contains $[G,G]$ is normal in $G$.
He says this as ...
52
votes
2
answers
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Yoneda-Lemma as generalization of Cayley`s theorem?
I came across the statement that Yoneda-lemma is a generalization of Cayley`s theorem which states, that every group is isomorphic to a group of permutations.
How exactly is Yoneda-lemma a ...
38
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9
answers
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Alternative proof that the parity of permutation is well defined?
I learned the following theorem about the properties of permutation from Gallian's Contemporary Abstract Algebra.
Theorem 5.4 $\;$ Always Even or Always Odd
If a permutation $\alpha$ can be ...
37
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1
answer
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"Efficient version" of Cayley's Theorem in Group Theory
I'm considering finite groups only.
Cayley's theorem says the a group $G$ is isomorphic to a subgroup of $S_{|G|}$.
I think it's interesting to ask for smaller values of $n$ for which $G$ is a ...
31
votes
1
answer
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Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups?
I think the answer is yes.
Sketch of the proof Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $\{e_\lambda:\lambda\in\Lambda\}\subset\mathbb{R}$ be its Hamel basis. Then $\{(e_{\...
30
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4
answers
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$(\mathbb{Q},+)$ has no maximal subgroups
I have a problem that I don't have any idea.
Show that group $(\mathbb{Q},+)$ has no maximal subgroups.
27
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Every group of order 255 is commutative
There was previous task was same but with $N = 185$. And I prove it by showing that number of Sylow subgroups is 1 for every prime $p\mid N$. But there I have some options $N_5 \in \{1, 51\}$, $N_{17} ...
25
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1
answer
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Group of order $pqr$, $p < q < r$ primes
Let $G$ be a group such that $|G|=pqr$, $p<q<r$ and $p,q,r$ are primes. i need to prove that:
There exists a subgroup $H$ such that $H\unlhd G$ and $|H|=qr$.
$G$ is solvable.
$r$-Sylow subgroup ...
25
votes
1
answer
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If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
Let $G$ be a group with $|G| = n$ and let $ \emptyset \ne S \subseteq G$.
I want to show that $S^n$ is a subgroup of $G$ where by $S^n$ I mean the set $\lbrace s_1\cdots s_n \; | \; s_i \in S\rbrace$...
20
votes
7
answers
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In what sense of "structure" do group homomorphisms "preserve structure"?
It is commonly said that group homomorphisms "preserve the structure of the group", e.g., from Wikipedia:
The purpose of defining a group homomorphism as it is, is to create functions that preserve ...
18
votes
3
answers
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Show that any abelian transitive subgroup of $S_n$ has order $n$
Can anybody tell me what is known about the classification of abelian transitive groups of the symmetric groups?
Let $G$ be a an abelian transitive subgroup of the symmetric group $S_n$. Show that $...
15
votes
5
answers
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A finite group which has a unique subgroup of order $d$ for each $d\mid n$.
Problem
Suppose G is a finite group of order $n$ which has a unique subgroup of order $d$ for each $d\mid n$. Prove that $G$ must be a cyclic group.
My idea: I try to prove it by induction. Let $p|...
15
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8
answers
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If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic
Fraleigh(7th ed) Sec10, Ex47. Let $G$ be a finite group. Show that if for each positive integer $m$ the number of solutions $x$ of the equation $x^m=e$ in $G$ is at most $m$, then $G$ is cyclic.
I ...
13
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4
answers
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Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions
I have some difficulties with a task in algebra. I guess it's trivial and really easy but I can't figure out how to solve it.
I have a set $G$ and a binary operation on it, let it be $\circ$. I have ...
13
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4
answers
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Any Set with Associativity, Left Identity, Left Inverse is a Group
Related Link: Right identity and Right inverse implies a group
Reference: Fraleigh p. 49 Question 4.38 in A First Course in Abstract Algebra
I will present my proof (distinct from those in the ...
13
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3
answers
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If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
This is not for homework. (I am a grader for a class.)
The case in which $G$ is finite is trivial. (That is, use a corollary to Lagrange's Theorem, and set $[G:H] = \dfrac{|G|}{|H|}$, and similarly ...
12
votes
3
answers
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When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
Suppose you have a group isomorphism given by the first isomorphism theorem:
$$G/\ker(\phi) \simeq \operatorname{im}(\phi)$$
What can we say about the group $\ker(\phi)\times \operatorname{im}(\phi)$? ...
9
votes
3
answers
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Cancellation of Direct Products
Given a finite group $G$ and its subgroups $H,K$ such that $$G \times H \cong G \times K$$ does it imply that $H=K$.
Clearly, one can see that this doesn't work out for all subgroups. Is there any ...
8
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3
answers
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Is every primitive element of a finite field of characteristic $2$, a generator of the multiplicative group?
Let $\alpha\in \overline {\mathbb F_2}$ (the algebraic closure of $\mathbb F_2$ ) be such that $\mathbb F_2[\alpha]$ is a field of order $2^n$ (where $n>1$).
Then is it true that $\alpha \in \...
5
votes
2
answers
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Classifying the factor group $(\mathbb{Z} \times \mathbb{Z})/\langle (2, 2) \rangle$
We wish to classify the factor group $(\mathbb{Z} \times \mathbb{Z})/\langle (2, 2) \rangle$, that is, find a group to which it is isomorphic. (According to the fundamental theorem of finitely ...
4
votes
2
answers
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Uniqueness follows from explicit solution of $\ a + x = b\,$ in a group
I just started real analysis. I don't have a background in proofs or logic, simply calculus. So I'm trying to learn more about proofs--so forgive the basic question, please.
How do you go about ...
4
votes
1
answer
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If $|A| > \frac{|G|}{2} $ then $AA = G $ [closed]
I'v found this proposition.
If $G$ is a finite group , $ A \subset G $ a subset and $|A| > \frac{|G|}{2} $ then $AA = G $.
Why this is true ?
4
votes
2
answers
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Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian [duplicate]
Possible Duplicate:
Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
If $G/Z(G)$ is cyclic, then $G$ is abelian
If $G$ is a group and $Z(G)$ the center of $G$, show that if $G/Z(...
2
votes
2
answers
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Proving Sylow's first theorem
Let $p$ be prime and $G$ a group such that $|G| = p^n k$, where $p \nmid k$. We want to show that $G$ has at least one Sylow $p$-subgroup.
Let $\mathcal{S} := \{ S \subset G \mid \, |S| = p^n\}, |\...
2
votes
2
answers
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Showing that $G$ is a group under an alternative operation.
Let $G$ be a group and let $c$ be a fixed elements of $G$.
Now, I'm going to define a new operation "*" on $G$ by $a*b=ac^{-1}b$
How do I prove that the set $G$ is a group under *.
Thanks for ...
102
votes
4
answers
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How is a group made up of simple groups?
I've read more than once the analogy between simple groups and prime numbers, stating that any group is built up from simple groups, like any number is built from prime numbers.
I've recently started ...
59
votes
1
answer
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Isomorphic quotients by isomorphic normal subgroups
In this recent question, Iota asked if, given a finite group $G$ and two isomorphic normal subgroups $H$ and $K$, it would follow that $G/H$ and $G/K$ are isomorphic. This is not true (a simple ...
55
votes
2
answers
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Can someone explain Cayley's Theorem step by step?
This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. ...
45
votes
2
answers
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Structure Theorem for abelian torsion groups that are not finitely generated?
I know about the structure theorem for finitely generated abelian groups.
I'm wondering whether there exists a similar structure theorem for abelian groups that are not finitely generated. In ...
36
votes
2
answers
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Showing that a finite abelian group has a subgroup of order $m$ for each divisor $m$ of $n$
I have made an attempt to prove that a finite abelian group of order $n$ has a subgroup of order $m$ for every divisor $m$ of $n$.
Specifically, I am asked to use a quotient group-induction argument ...
33
votes
4
answers
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What is $\gcd(0,a)$, where $a$ is a positive integer?
I have tried $\gcd(0,8)$ in a lot of online gcd (or hcf) calculators, but some say $\gcd(0,8)=0$, some other gives $\gcd(0,8)=8$ and some others give $\gcd(0,8)=1$. So really which one of these is ...
28
votes
4
answers
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Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.
Is my proof below correct? What specific property of rationals did I exploit in my proof? It looks like the property I exploited is the following: Given any positive rational, I can always write it as ...
23
votes
5
answers
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$\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$
I'd like to see a proof why $\varphi \in \operatorname{Hom}{(S^1, S^1)}$ looks like $z^n$ for an integer $n$.
At first I thought I could argue that if I have a homomorphism that maps $e^{ix}$ to some ...
22
votes
2
answers
30k
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Painting the faces of a cube with distinct colours
I don't think this is solved by Burnside's Lemma since there is a condition that each side is painted a different colour. The question is as follows.
If I had a cube and six colours, and painted ...
21
votes
2
answers
2k
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Has this "generalized semidirect product" been studied?
If $G$ is a finite group with subgroups $H$ and $K$ such that $HK = G$ and $H\cap K = \{1\}$ we get that every element of $G$ can be written uniquely as $hk$ with $h\in H$ and $k\in K$. This then ...
18
votes
2
answers
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Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$, $\forall x\in G$. Prove that $H$ is a normal subgroup of $G$.
I have tried to using the definition but failed. Can someone help me please.
18
votes
2
answers
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For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
I recently read the well known theorem that for a group $G$ and $H$ a normal subgroup of $G$, then $G$ is solvable if and only if $H$ and $G/H$ are solvable. In my book, only the fact that $G$ is ...
17
votes
1
answer
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Converse of Lagrange's theorem for abelian groups
I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup ...