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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Order of a certain finitely generated group

Suppose I am looking at the group $W=\langle s_\alpha, t_\beta \rangle$ where $s_\alpha$ and $t_\beta$ are reflections in $\mathbb{R}^2$ coming from two vectors $\alpha$ and $\beta$ making an angle of ...
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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 ...
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6answers
79 views

Prove or give a counterexample: If a group G has a subgroup of order n, then G has an element of order n

The question is given in the title. I am unable to come up with a counterexample and I'm thinking this could apply to cyclic groups but not necessarily to general groups. Does anyone have any ideas or ...
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1answer
33 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 ...
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1answer
39 views

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

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 ...
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1answer
31 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 ...
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1answer
22 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 ...
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1answer
52 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 $\quad$ $A\times (B \times C) \cong (A\times B) \times C$; $\quad$ $1 \times A \cong A \cong A\times 1$; ...
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3answers
53 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)) ...
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0answers
22 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) ...
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1answer
33 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 ...
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1answer
27 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) ...
2
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0answers
28 views

Number of elements of order $6$ in $\text{Aut}(\mathbb Z_{720})$

I tried to determine the number of elements of order $6$ in $\text{Aut}(\mathbb Z_{720})$. Please could someone tell me if this is correct? $$ \text{Aut}(\mathbb Z_{720}) \cong U(720) \cong U(9) ...
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1answer
42 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 $G$ ...
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1answer
22 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 ...
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1answer
51 views

Group of order 6 contains an element of of order 3

I need to show that if $G$ is non-abelian group of order $6$ then it contains an element of order $3$. I don't know how to proceed. Any kind of help/hint is appreciated.
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1answer
46 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 $D_{15}$ ...
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1answer
43 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 ...
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1answer
49 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, ...
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0answers
41 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 $n = \#A$. ...
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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 ...
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1answer
24 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 ...
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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 ...
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1answer
17 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 ...
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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 ...
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1answer
31 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 ...
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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: ...
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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 ...
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0answers
20 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 ...
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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 ...
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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 ...
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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 $1$ : ...
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40 views

How to find out generators of the following free group?

The following is the quotient group of SL($2,\mathbb{Z}$). Consider $(H/\{-1,1\} \cap H)$ where $H=\begin{bmatrix}2\mathbb{Z}+1&4\mathbb{Z}\\2\mathbb{Z}&2\mathbb{Z}+1\end{bmatrix}$ How do ...
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2answers
29 views

how to find the index of following subgroup?

if I denotes the principal congurence group of level 2 i.e. $I=\{ M \in SL(2,Z) ; \:M \:\:\text{congruent to I} \mod(2)\}$. or I= ...
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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}, ...
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2answers
43 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 ...
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1answer
41 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 ...
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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?
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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 $H$ ...
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All equivalent moves on a rubik's cube

Call the primitive moves on the rubik's cube "R,L,U,D,F,B" for right, left, up, down, front, and back respectively. Let us say that I have a permutation of the stickers on the cube written as a word ...
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1answer
34 views

Where can I find a proof of the Presentation for Semidirect Products?

I have seen it claimed online that: Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\theta \colon H \to \text{Aut}(G)$, then $$ G\rtimes_\phi ...
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1answer
57 views

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

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, ...
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0answers
29 views

Let $X : S_3 → GL_2(\mathbb{R})$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group ...
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1answer
22 views

Several true/false statements about a finite group $a,g\in G$ such that $a$ is of order $2$

Let $G$ be a finite group, and $a,g\in G$ such that $a$ is of order $2$, then the following is either true or false: The element $gag^{-1}$ is of order $2$. $(ag)^2=g^2$ if $ag$ is of ...
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1answer
49 views

Order and generators of intersection of cyclic groups

Let $\sigma$, $\tau$ be two permutations of $S_n$. We know that $\langle \sigma \rangle \cap \langle \tau \rangle$ is a cyclic subgroup and (from Lagrange's Theorem) that its order divides ...
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2answers
14 views

Multiplying elements of a multiplicative cyclic group

This is about correct operation in a multiplicative cyclic group. Say we have the group $\mathbb Z_4^{\times} = \{1,2,3\}$, if we multiply the elements, say $3\cdot 2\cdot 2 $ we get $12 \mod 4 = 0$ ...
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1answer
37 views

Characteristic subgroups of order $2$

Could anybody give an example of a finite abelian $2$-group with more than one characteristic subgroup of order $2$ ? (In other words, a finite abelian $2$-group $G$ with a Klein subgroup $V$ such ...
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29 views

Estimate the number of “solvable” polynomials with degree $d$ and coefficient limit $l$

Enumerate the irreducible polynomials with degree $d$ and integer coefficients $[-l,l]$ and check how many of them have a solvable galois-group. We can assume that the leading coefficient of the ...
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1answer
15 views

normaliser of dihedral group of order 4

Let $D_{2n}$ be the dihedral group of order $2n$. I would like to show that if $4$ divides $n$, then the normaliser $N_{D_{2n}}(D_{4})=D_{8}$. I know that ...
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3answers
60 views

$G$ is a finite group, if $m\in \mathbb N$ such that $g^m=1$ for all $g\in G$, then $m=n$.

Prove / disprove: Let $G$ be a finite group of order $n$, if $m\in \mathbb N$ such that $g^m=1$ for all $g\in G$, then $m=n$. I can't find a group that will satisfy this condition so I think it's ...