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.

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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 ...
Vladimir's user avatar
  • 2,879
19 votes
3 answers
2k views

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 ...
misi's user avatar
  • 997
17 votes
3 answers
11k views

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?
nomadicmathematician's user avatar
15 votes
3 answers
5k views

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 ...
HaroldF's user avatar
  • 1,510
14 votes
4 answers
3k views

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 (...
Leo's user avatar
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13 votes
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.
Shiva Prakash's user avatar
13 votes
3 answers
6k views

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, ...
Pax's user avatar
  • 5,762
12 votes
2 answers
10k views

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 $\...
user77404's user avatar
  • 1,177
9 votes
3 answers
6k views

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 ...
NebulousReveal's user avatar
7 votes
4 answers
9k views

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 ...
Tumbleweed's user avatar
  • 1,001
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 ...
Mr Bingley's user avatar
3 votes
5 answers
8k views

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 $\text{ord}(x) = \text{ord}(x')$? ...
Littletry's user avatar
68 votes
4 answers
13k views

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 ...
Bruno Stonek's user avatar
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52 votes
2 answers
7k views

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 ...
Jan's user avatar
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38 votes
9 answers
11k views

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 ...
user avatar
37 votes
1 answer
1k views

"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 ...
Rose's user avatar
  • 371
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 $\{(e_{\...
Norbert's user avatar
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30 votes
4 answers
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$(\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.
Muniain's user avatar
  • 1,453
27 votes
7 answers
9k views

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} ...
RiaD's user avatar
  • 1,272
25 votes
1 answer
17k views

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 ...
giladude's user avatar
  • 963
25 votes
1 answer
2k views

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 $\lbrace s_1\cdots s_n \; | \; s_i \in S\rbrace$...
Robert M's user avatar
  • 1,558
20 votes
7 answers
6k views

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 ...
Dennis's user avatar
  • 2,517
18 votes
3 answers
4k views

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 $...
abe's user avatar
  • 181
15 votes
5 answers
11k views

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|...
Hang's user avatar
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15 votes
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 ...
Gobi's user avatar
  • 7,438
13 votes
4 answers
7k views

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 ...
Faery's user avatar
  • 283
13 votes
4 answers
12k views

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 ...
Andy Tam's user avatar
  • 3,357
13 votes
3 answers
7k views

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 ...
Clarinetist's user avatar
  • 19.5k
12 votes
3 answers
3k views

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 $\ker(\phi)\times \operatorname{im}(\phi)$? ...
user24273's user avatar
  • 475
9 votes
3 answers
3k views

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 ...
user avatar
8 votes
3 answers
1k views

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 \...
user521337's user avatar
  • 3,705
5 votes
2 answers
4k views

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 ...
Andrew Thompson's user avatar
4 votes
2 answers
9k views

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 ...
rsteckly's user avatar
  • 143
4 votes
1 answer
776 views

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 ?
WLOG's user avatar
  • 11.4k
4 votes
2 answers
15k views

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(...
Lily's user avatar
  • 301
2 votes
2 answers
729 views

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\}, |\...
Huy's user avatar
  • 6,674
2 votes
2 answers
1k views

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 ...
user avatar
102 votes
4 answers
8k views

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 ...
Bruno Stonek's user avatar
  • 12.5k
59 votes
1 answer
11k views

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 ...
Arturo Magidin's user avatar
55 votes
2 answers
31k views

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 $\lambda_x (g) = xg$. ...
user72625's user avatar
  • 1,219
45 votes
2 answers
7k views

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 ...
Seoral's user avatar
  • 961
36 votes
2 answers
18k views

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 ...
Alex Petzke's user avatar
  • 8,763
33 votes
4 answers
53k views

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 ...
Vafa Khalighi's user avatar
28 votes
4 answers
40k views

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 ...
John Smith's user avatar
  • 1,869
23 votes
5 answers
3k views

$\varphi$ 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 ...
Rudy the Reindeer's user avatar
22 votes
2 answers
30k views

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 ...
Sputnik's user avatar
  • 3,764
21 votes
2 answers
2k views

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 ...
Tobias Kildetoft's user avatar
18 votes
2 answers
14k views

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.
hamidi's user avatar
  • 181
18 votes
2 answers
6k views

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 ...
yunone's user avatar
  • 22.3k
17 votes
1 answer
12k views

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 ...
saurs's user avatar
  • 1,377