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.

learn more… | top users | synonyms (2)

-1
votes
3answers
131 views

Only two groups of order $10$: $C_{10}$ and $D_{10}$

Show that up to isomorphism there exist only two groups of order 10: $C_{10}$ and $D_{10}$. I need some help on this question. I only know the basic definitions of isomorphism. Any hints?
0
votes
0answers
6 views

Generating space representation of all elements of a point group using generators

A well known set of groups are the 32 three-dimensional crystallographic point groups which elements represent the transformation matrices of the symmetry elements (rotation, reflection etc.). If one ...
0
votes
1answer
23 views

$G$ is a finite group, if $ab^{-1}=ba^{-1}$ and $n$ is odd, then $a=b$

$G$ is a finite group of order $n$, then if $a,b\in G : ab^{-1}=ba^{-1}$ and $n$ is odd, then $a=b$. multiply both sides by $ab^{-1}$ we get $(ab^{-1})^2 = ab^{-1}ab^{-1}=ba^{-1}ab^{-1}=1$ so ...
-2
votes
1answer
38 views

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R\{1}. [on hold]

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R \ {1}. (a) Show that a ∗ b ∈ G for all a, b ∈ G. (b) Show that G together with the binary operation G × G → G, (a, b) |→ a ...
0
votes
1answer
26 views

Number of permutations with given cyclic structure

If $\sigma$ is a permutation made up by the disjoint cycles $\tau_1, \dots, \tau_r$ (including those of length $1$), we call structure of $\sigma$ $$(l_1, \dots, l_r),$$ where $l_1, \dots, l_r$ are ...
8
votes
3answers
430 views

For what $n$ is $U_n$ cyclic?

When can we say a multiplicative group of integers modulo $n$, i.e., $U_n$ is cyclic? $$U_n=\{a \in\mathbb Z_n \mid \gcd(a,n)=1 \}$$ I searched the internet but did not get a clear idea.
0
votes
0answers
18 views

Structure of frobenius groups.

Definition- A groups $G$ is called Frobenius if it has a proper nontrivial subgroup $H$ such that $H \cap H^g=1\ \forall\ g\in G-H$. Do we have a structure or classification theorem for (finite) ...
2
votes
1answer
44 views

Can we define structures like groups or monoids in the context of pure category theory?

In a category $\mathcal C$ with terminal object $1$ and objects $A$, $B$, $C$ we have $\quad$ $A\times (B \times C) \cong (A\times B) \times C$; $\quad$ $1 \times A \cong A \cong A\times 1$; ...
3
votes
3answers
50 views

Find all homomorphisms $Q \rightarrow \mathbb{Z}_8$

Let $Q$ be the quaternion group. Find all homomorphisms $\phi: Q \rightarrow \mathbb{Z}_8$ What I get into is one big ifology: Of course $\phi(1) = 0$, then $0 = \phi(1) = \phi(-1 \cdot (-1)) ...
0
votes
1answer
20 views

Verify $(H_1 \cap H_2)g=H_1g \cap H_2g$ where $H_1,H_2\leq G$ and $g\in G$.

Verify $(H_1 \cap H_2)g=H_1g \cap H_2g$ where $H_1,H_2\leq G$ and $g\in G$. Let $S_1,S_2\leq G,g\in G$ What i had done, $x\in (S_1 \cap S_2)g$. Then $x=sg,s\in S_1\cap S_2$. So clearly, $x\in ...
1
vote
1answer
45 views

how to find coefficient c1, c2, c3, c4 of a polynomials of degree 4 from resolvent

if not starting from standard resolvent of each degree and use (y-x1...)(y-x2...)(y-x3...) and group theory how to find corresponding c1, c2, c3, c4 of polynomial x^4+c4*x^3+c3*x^2+c2*x+c1 which c1, ...
2
votes
1answer
29 views

What am I missing here: $U(144) \neq U(140)$

I'm confused about the following exercise: Prove that $U(144)$ is isomorphic to $U(140)$. Here are my thoughts: $$U(144) = U(12^2) = U(3^2)\oplus U(2^4) = \mathbb Z_{6} \oplus \mathbb Z_{8}$$ and ...
1
vote
1answer
22 views

Why $U(p^m) \oplus U(q^n)$ is not cyclic

I tried to solve the following exercise: Let $p,q$ be odd primes and $n,m$ positive integers. Explain why $U(p^m) \oplus U(q^n)$ is not cyclic. I solved the question as follows: We have $U(p^m) ...
5
votes
1answer
38 views

How to Identify a Quotient of a Given Free Group

$\newcommand{\Z}{\mathbf Z}$ Problem. Let $G$ be the free group generated by three symbols $a, b$ and $c$, and let denote $G$ by writing $F(a, b, c)$. Let $N$ be the normal subgroup of $G$ ...
0
votes
0answers
39 views

How to find a onto homomorphism between two groups?

Consider the following subgroups of $\text{SL}(2,\mathbb{Z})$ : $A$ the subgroup of matrices with determinant $1$ : ...
1
vote
1answer
42 views

Determine all the subgroups of the dihedral group $D_{15}$

Is there an algorithm for finding all of the subgroups of $D_{15}$? Also, is there a formula for finding the size of that subgroup? Not sure where to start with finding all the subgroups of $D_{15}$ ...
2
votes
1answer
21 views

Action via automorphism

I want to ask what does it mean to say a group $A$ acts on $N$ via automorphisms. It is a notion used in M.Isaacs book and I am not familiar with. I tried to find how it is defined but a scanned e ...
-1
votes
1answer
40 views

showing non-isomorphism of groups [duplicate]

How do I prove that there is no isomorphism between $\Bbb Z$ under addition and $\Bbb Q$ under addition? They both are infinite order. I thought they might be isomorphic. Help would be ...
1
vote
2answers
174 views

How do I show that two groups are not isomorphic?

Prove that the additive groups of $\Bbb Z$ and $\Bbb Q$ are not isomorphic. It is hard to create a map that will show homomorphism.
26
votes
1answer
385 views

Are there $n$ groups of order $n$ for some $n>1$?

Denote $N(n)$ : the number of groups with order $n$. Can $N(n)=n$ hold for some $n>1$ ? I checked the OEIS-sequence as well as the squarefree numbers $n$ in the range $[2,10^6]$ and found no ...
0
votes
1answer
43 views

Groups isomorphic to $S_{4}/N$

Let $G = S_4$ be a group, $N = \{1, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\}$ a normal subgroup of G. It's easy to see that $G/N$, the set of cosets is $G/N = \{a, b, c\}$, where $$a = \{(1), (1, ...
2
votes
0answers
38 views

Finding homomorphisms in normal groups to $S_n$

This question is related to my trouble in abstract algebra in general so some general advice is really appreciated!! (I'm really new to this.) Let A be an infinite subset of G with $n = \#A$. ...
2
votes
3answers
31 views

Two disjunct normal subgroups

Let M, N be normal subgroups of G with M∩N={e}. I'm trying to prove that MxN is isomorphic to G. I proved that nm=mn for all n in N and m in M. So now I'm trying to take any fixed g in G and ...
0
votes
1answer
23 views

Structure of induced module for subgroup $H \le G$, confused by mixing up notions of $FH$-submodule and $F$-subspace

The following is a question on a passage in the book A Course in the Theory of Groups by Derek Robinson. There he constructs the induced module $M^G$ from a $FH$-module for some subgroup $H \le G$ of ...
1
vote
2answers
45 views

Showing a non-isomorphism of groups

I need to show that $\Bbb Z^*_8$ is not isomorphic to $\Bbb Z^*_{10}$. $\Bbb Z^*_n$ means integers up to $n$ coprime with $n$ I do not know how to do this. I have difficulties doing proofs ...
2
votes
0answers
99 views

Number of non-isomorphic groups of square-free order $n$

$G$ is a group of order $n=p_1p_2....p_k$ then how to find the number of non-isomorphic groups of order $n$ where $p_i's$ are distinct primes I can find the number of number of non-isomorphic ...
0
votes
1answer
83 views

Number of non-isomorphic groups of order 21

Let $G$ be a group of order $21$. Find the number of non-isomorphic groups of order $21$ My solution: If the group $G$ is commutative,then $G$ can be expressed as a direct product of cyclic groups ...
1
vote
1answer
16 views

Finding all permutations which satisfy given condition

In a symmetric group $S_n$ find number of permutations $P$ such that in the disjoint cycle decomposition of $P$ , length of cycle containing $1$ is $k$ . Here's my attempt at this . I found number of ...
0
votes
1answer
30 views

homomorphism of profinite groups

Let $G$ be a profinite group and consider a continuous surjective homomorphism: $$\phi:G\rightarrow \widehat{\mathbb Z}$$ where $\widehat{\mathbb Z}:=\varprojlim \mathbb Z/n\mathbb Z$. Moreover Let ...
1
vote
1answer
12 views

order of a subrgoup of rank $r\geq 2$ in $\mathbb{F}_p^*$

Let $a,b\in \mathbb{F}_p^*$ with orders $o_p(a)=|\langle a \rangle|=\alpha$ and $o_p(b)=|\langle b \rangle|=\beta$. I have few questions: 1) Is it true in this case ($\mathbb{F}_p^*$ cyclic) that ...
1
vote
0answers
31 views

On the probability distribution of iterated permutations

I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it: ...
0
votes
0answers
22 views

How can I define $H+K$? [duplicate]

Let be integers 5 and 100, and let be $H=5Z$ and $K=100Z$ subgroups of the additive group $Z$. How can I define the subgroup $H+K$ ? I think $5Z+100Z=5Z$ because mcd(100,5)=5 but I'm not sure that ...
2
votes
3answers
1k views

How to determine the number of isomorphism types of groups of a given order?

if $G$ is a group whose order is $n$ can we determine the number of isomorphism types for this number or not ? for instance, if $n=4$ we have 2 types, $Z_4$ and $Z_2 \times Z_2$ " Klein 4-group" ...
3
votes
1answer
75 views

Are there cube-free numbers $n$, for which the number of groups of order $n$ is unknown?

For squarefree $n$, there is a formula allowing to compute the number of groups of order $n$. I do not think that such a formula exists for cubefree numbers. If a cubefree number $n$ has the ...
4
votes
1answer
143 views

Is $n=8{,}574{,}796{,}230$ the smallest squarefree number $n$ with $gnu(n)>10^6$?

The number of groups of order $n$ (gnu(n)) can be calculated by a closed formula, if $n$ is squarefree. I could calculate the values with GAP, additionally, I programmed a version in PARI/GP, working ...
3
votes
1answer
39 views

In proof of $|G| = n_1^2 + n_2^2 + \ldots + n_h^2$ how could equality of dimensions concluded from ring isomorphism

In Derek Robinson, A Course in the Theory of Groups on page 224 he proves: Let $G$ be a finite group and let $F$ be an algebraically closed field whose characteristic does not divide the order of ...
0
votes
0answers
18 views

Conjugate classes of Symmetric group, $S_n$ and partition number of $n$.

In the book "Topics in Algebra", 2nd edition, By I.N. Herstein, the following lemma is given on page 89, Lemma 2.11.3 :The number of conjugate classes in $S_n$(the symmetric group of order $n$) is ...
2
votes
0answers
23 views

About a perfect FN-group

the consept perfect group is new for me. I was reading the following: If a group $G$ is a FN-group and $G'=G$ then $G$ is finite. I tried to prove this and here is my attempt: let $G$ be a ...
1
vote
1answer
48 views

Determine if quotient group $S_4/N$ is isomophic to $S_3$ [duplicate]

Let $N = \{1,(12)(34),(13)(24),(14)(23)\}$. Determine if the quotient group $S_4/N$ is isomophic to $S_3$. I computed the cosets: $N, (12)N, (13)N, (14)N, (123)N, (234)N$, and the others are ...
-2
votes
1answer
59 views

Is the symmetric group $S_4$ cyclic

Is the symmetric group $S_4$ cyclic? By writing all $24$ elements we can write the tabular form of $S_4$. Then choosing each element of $S_4$, we can find its order and thus, we can show that ...
3
votes
4answers
579 views

Prove that the symmetric group $S_n$, $n \geq 3$, has trivial center.

I am trying to prove this: Let $\sigma$ be a non-identity element of $S_{n}$. If $n \geq 3$ show that $\exists \gamma \in S_{n}$ such that $\sigma\gamma \neq \gamma\sigma$. Hint: Let ...
3
votes
4answers
613 views

Are there any Symmetric Groups that are cyclic?

Are there any Symmetric Groups that are cyclic? Because I have been doing some problems and I tend to notice that the problems I do that involve the symmetric group are not cyclic meaning they do not ...
1
vote
1answer
40 views

Prove that $g\in P$

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup. If $g$ is an element of $G$ with order a power of $2$ then it lies in some conjugate of $P$. I get this. But if also $g\in C_G(P)$, then it is ...
0
votes
1answer
32 views

Which of the following are isomorphic?

I am a beginning learner of group theory. Which of the following are isomorphic:$$\mathbb{Z_{24}}, D_{4}\times \mathbb{Z_{3}},A_{4}\times \mathbb{Z_{2}},\mathbb{Z_{2}}\times D_{6}, ...
-1
votes
1answer
50 views

Subgroups of $G = \{\sigma \in S_{14}: \mathrm{ord}(\sigma)\mid 12 \}$ [on hold]

Let $$G = \{\sigma \in S_{14}: \mathrm{ord}(\sigma)\mid 12 \}.$$ Is this a subgroup? By examining particular examples, one can see that it is not, since it is not closed under composition. However, ...
2
votes
2answers
41 views

Morphisms of a category with one object, which is a group

I've read two articles(1,2), but still have a question about structure of group, say $G$, as an one-object category. Let's call this category $C$. I understand that morphisms of $G$, which is ...
3
votes
1answer
391 views

Groups of order $pq$ have a proper normal subgroup

I am doing the following exercise from [Birkhoff and MacLane, A survey of modern algebra]: Let $G$ be a group of order $pq$ ($p,q$ primes). Show that either $G$ is cyclic or contains an element ...
1
vote
1answer
39 views

Expressing $z\in G$ as $z=gh^2$ where $g$ is a $2$-element and $|h|$ is odd

Let $z\in G$ where $G$ is a finite group, then is it always true that there exist elements $g,h\in G$ such that $z=gh^2$ where $|g|=2^k$ for $k \in \Bbb{Z_{\ge0}}$ and $|h|$ is odd?
1
vote
2answers
26 views

reference request Schur Zassenhaus Theorem

I am looking for a reference for the Schur Zassenhaus Theorem, saying that any normal Hall subgroup admits a complement. An on-line search show that it is supposed to be in "The theory of groups" by ...
0
votes
1answer
37 views

Showing a subgroup contains the identity element

Let $G$ be the set of all functions from $\mathbb{R}$ to $\mathbb{R}$ along with $+$. Show that $H$ defined by $H=\{f:f(x)=0 \text{ for all } x \in [0,1]\}$ is a subgroup. I am able to show $H$ ...