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)

2
votes
2answers
56 views

Find the order of the elements in the given groups

I have to find the order of the following elements in the given groups: $(1 \ \ 2 \ \ 3) \ (1 \ \ 2\ \ 4) \text{ in } S_5$ $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 3 & 4 & 1 ...
2
votes
1answer
123 views

the number of all orders of elements in HS (Higman-Sims group) with GAP

I want to calculate the number of all orders of elements in HS (Higman-Sims sporadic simple group). Is there any way of doing this with MAGMA or GAP? How I can determine orders of elements of a group ...
2
votes
1answer
110 views

Show that there are infinitely many primes $p$ such that $p = 1 (\mod q)$ in a very specific way

I friend of mine has shown the following: Let $n \in \mathbb{N}$, and $q$ an odd prime number. Any $p$ dividing $1 + n + \cdots + n^{q-1}$ satisfies $p \equiv 1 (\mod q)$, whenever $ n \not \equiv ...
2
votes
1answer
58 views

Direct limit of subgroups

Let $G$ be a group and $G^i$ a collection of subgroups which form a direct system over a directed set $I$, so $i\leq j \iff \exists\; \varphi^i_j: G^i\to G^j$ where $\varphi^i_j$ is the inclusion map. ...
2
votes
1answer
74 views

The Trivial Centre of Group $G/Z(G)$ [closed]

Let $G$ be a group such that $G = G'$, where $G'= [G,G]$ is the Derived Subgroup of $G$. Prove that the centre o group $G/Z(G)$ is trivial, that is, prove that $Z\left(G/Z(G) \right) = 1$.
2
votes
1answer
170 views

Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
2
votes
1answer
83 views

Presentation of a group: Show that $\langle a|a^2\rangle =\{1,a\}$.

Example: $\langle a|a^2\rangle=\{1,a\}$. After reading the definition of presentation of a group, I find myself cannot understand the above example given. I don't know which part of the definition I ...
2
votes
1answer
33 views

Short exact sequence with binary tetrahedral group does not split

The following is a short exact sequence, where $T$ is the binary tetrahedral group (equivalently the Hurwitz units), and $Q$ is the quotient of $T$ by $\mathbb{Z}/2$. $1 \rightarrow \mathbb{Z}/2 ...
2
votes
2answers
92 views

Group theory and group representation

I am fairly new to group theory and representation. I am currently looking at faithful representations. I am not quite sure what is the "use" of a faithful representation. I cannot find any "easy to ...
2
votes
1answer
67 views

Find the Automotphism group of direct product of Z(mod m) & Z(mod n).

Find the Automotphism group of direct product of $\mathbb{Z}(\operatorname{mod} m)$ and $\mathbb{Z}(\operatorname{mod} n)$. We know Automorphism of direct product of $\mathbb{Z}(\operatorname{mod} ...
2
votes
1answer
67 views

Why the paired orbit has the same size here?

enter link description here On this proof, he just showed that the the paired orbit of (a,b) has the same size, but this seem has nothing to do with the size of corresponding paired orbit of an orbit ...
2
votes
1answer
82 views

Number of Sylow 2-subgroups of a special linear group

Find the number of Sylow $2$-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo $3$). I think it will be $1$. But I failed to prove it using the counting principle. It has $4$ ...
2
votes
1answer
75 views

Representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely ...
2
votes
1answer
64 views

Number of actions of $\mathbb Z$

Let $X$ be a finite set. Determine the number of actions of $\mathbb Z$ on $X$. If $X$ is a finite set with $|X|=m$, then $|\{f:X \to X : \text{f is bijective}\}|=m!$. Finding the number of actions ...
2
votes
1answer
219 views

Infinite group having composition series

Example of an infinite group having composition series. I have examples as infinite alternating group and projective special linear group. But I want example other than infinite simple group.
2
votes
2answers
61 views

A question on the order of an element involving relatively primes

This question is based on an exercise that comes from the second chapter of Malik's Fundamentals of abstract algebra which states as follows (I paraphrase): Let $(G, *)$ be a group and $x\in G$. ...
2
votes
1answer
67 views

What can you say about the order of this element?

I am self-studying algebra and encountered the following problem: If $b^{-1}ab=a^2$ and $c^{-1}ac=a^3$ and $b,c$ has orders $4$ and $3$ respectively, what can you say about the order of $a$? In ...
2
votes
1answer
272 views

every Abelian group is a converse lagrange theorem group

Let $G$ be a finite abelian group, then $G$ has a subgroup of order $n$ if and only if $n\mid G$. Proof: by Lagrange if $H\leq G$ then $|H|$ divides $|G|$ so this proves one of the implications. We ...
2
votes
1answer
105 views

The smallest composite integer $n$ such that there is a unique group of order $n$?

We need to find the smallest composite integer $n$ such that there is a unique group of order $n$. Attempt: Let us suppose that $n= ab$ is a composite integer where $a,b$ are integers such that ...
2
votes
1answer
67 views

$S=\{a\in G:f(x)=f(ax)\; \forall x\in G\}$ is a subgroup of $G$

Let $G$ be a group and $f:G\rightarrow G$ a function. Let $S=\{a\in G:f(x)=f(ax)\; \forall x\in G\}$. Prove that $S$ is a subgroup of $G$. This is my first encounter with functions in this ...
2
votes
1answer
65 views

Is there a concise argument for why this group operation is associative?

Given disjoint groups $(G,\cdot)$ and $(H,\ast)$ and an isomorphism $f:G\to H$, I've been able to show that the binary operation $\diamond$ on $G\cup H$ defined by $a\diamond b = a\cdot b$ if $a,b ...
2
votes
1answer
85 views

Constructing a group with given normal subgroups.

Let $N_1,N_2,\dots,N_n$ be simple groups. Is there is a group $G$ with exactly $n$ nontrivial proper normal subgroups isomorphic to $N_1,N_2,\dots,N_n$?
2
votes
1answer
114 views

Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That ...
2
votes
1answer
39 views

List of groups with specific divisors

I want to find list of finite simple nonabelian groups which their orders divisor lies in specific set of primes for example the set $\{2,3,5,7,11\}$. Is there a method to do this? Would Zsigmondy's ...
2
votes
1answer
63 views

Extension theory and automorphism extension

My question is motivated by the following two posts On finite 2-groups that whose center is not cyclic and Automorphisms of group extensions Question: Assume that $A,B,C$ are there algebraic ...
2
votes
1answer
60 views

Groups with order $p^3$ ($p$ prime) have two non commutative isomorphism classes

I read in an exercise, that a group with $p^3$ ($p$ prime) elements have $2$ non commutative isomorphism classes. Unfortunately there was just this statement without any explanation. We just solved it ...
2
votes
1answer
45 views

Relative orders of an element with respect to a subgroup

There is a theorem in an old monograph: Theorem 1. A pair of subgroups $A$ and $B$ forms a distributive pair if and only if for every element $c$ of $A\vee B$, not in $A$ nor in $B$, its relative ...
2
votes
2answers
42 views

The set of one-parameter subgroup of the Multiplicative group $G_m$ is Z

Let $G_m= k^{*}=k-{0}$ be the multiplicative group. We know this is an Algebraic group also. How does one prove any algebraic group morphism $G_m \rightarrow G_m$ is of the form $t \mapsto t^{n}$ for ...
2
votes
2answers
96 views

Cyclic subgroups of semidirect products

Let $H$ and $N$ be two finite groups and $\phi:H\to Aut(N)$ a homomorphism. Let $G=N\rtimes_\phi H$ and let $\pi:G\to G/N=H$ be the quotient map. Let $p$ be a prime which does not divide $|N|$ but ...
2
votes
1answer
223 views

Minimum number of out-shuffles required to get back to the start in a pack of $2n$ cards?

So I'm stuck on this problem. If you perform a faro out-shuffle (i.e. a perfect "riffle shuffle" where the top and bottom cards stays in place) on a pack of 52 cards ($n=26$), you can get back the ...
2
votes
1answer
79 views

Calculating sign of a permutation

Here's the permutation: $\pi\sigma= \left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 5 & 6 & 2 & 9 & 1 & 7 & 8 & 3 & 4 ...
2
votes
2answers
59 views

How to describe $G/U$?

Let $G=SL_2(\mathbb{C})$ and let $U = \{\left( \begin{matrix} 1 & x \\ 0 & 1 \end{matrix} \right): x \in \mathbb{C}\}$. We have an action of $U$ on $G$ by right multiplication. By definition, ...
2
votes
1answer
131 views

Projective special linear groups are co-hopfian?

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
2
votes
1answer
146 views

$\operatorname{Aut}(S_4)$ is isomorphic to $S_4$

I already proved this, but I think I can reduce my solution. My solution : There are 4 Sylow 3-subgroup of $S_4$, and denote the set of Syl 3-subgroups by $P=\{P_1,P_2,P_3,P_4\}$. Then, by a group ...
2
votes
1answer
57 views

Lower central series and the center of $S/S^k$

Let $S$ be a $p$-group and assume that $S$ has maximal class - so if we assume $\vert S \vert = p^n$ then the lower (and upper) central series has length $n-1$. I don't know much about lower central ...
2
votes
1answer
109 views

Trivial elements in $T(a,b,c)$

Consider the group $T(a,b,2)=<x,y|x^a, y^b, (xy)^2>$ and assume none of $a$ or $b$ is equal to $2$. How can one list all the trivial words (say up to length $11$ and apart from $(xy)^{2n})$) in ...
2
votes
1answer
74 views

Write cyclic groups of order $p^n$ in terms of simple groups

Some say that studying simple groups helps you understand the structure of non-simple groups. How can I write in terms of simple groups $\mathbb{Z}_{p^n}$? Eg. $\mathbb{Z}_9$
2
votes
2answers
211 views

How to count the number of elements of given order?

I am trying to prove the following result. Let $G$ and $G'$ be two finite abelian groups. Besides, they have the same number of elements of any given order. Prove that $G\cong G'$. My attempt is ...
2
votes
1answer
106 views

Subgroup contained in all other subgroups

This is Problem 2.13.10 from Herstein, Topics in Algebra: Let $G$ be a finite abelian group such that it contains a subgroup $H_0 \neq (e)$ which lies in every subgroup $H\neq (e)$. Prove that $G$ ...
2
votes
2answers
96 views

I.N. Herstein, “Topics in algebra” group theory section 2.8 example 2.8.1

I.N. Herstein, "Topics in algebra" group theory section 2.8 example 2.8.1 it is written that Let $G$ be a finite cyclic group of order $r$, $G=(a)$, $a^r=e$. Suppose $T$ is an automorphism of $G$. If ...
2
votes
1answer
105 views

Describe all group homomorphisms from $\mathbb Z$ to $D_8$

Describe all group homomorphisms from $\mathbb Z$ to $D_8$. So $\mathbb Z=\langle1\rangle$, and $f(1)$ determines the homomorphism. So every homomorphism is determined by $f(1)=\sigma$, and for some ...
2
votes
1answer
45 views

Minimal normal subgroups in a non-torsion group

Is there a group $G$ with an element with infinite order such that every non-trivial $N \unlhd G$ contains a minimal (non-trivial) normal subgroup of $G$?
2
votes
1answer
107 views

Decomposing direct product of irreps

I know characters of two 2-dimentional irreps (U and V) of a group with 6 conjugate classes. The characters are: $\begin{pmatrix} 2&-1&-1&2&0&0\end{pmatrix}$ and $\begin{pmatrix} ...
2
votes
2answers
43 views

Part of Proof which implies normal complement

"Let $N := O^{2'}(G)$, then $N \unlhd G$ is a normal subgroup, and suppose that $C_G(N) = G$. Then $G$ has a central Sylow $2$-subgroup. So $G$ has a normal $2$-complement." I do not understand the ...
2
votes
1answer
50 views

Is the intersection of distinct conjugations of Sylow subgroups trivial?

Let $G$ be a finite group. Let $P,Q$ be Sylow p-subgroups of $G$. If $P\neq Q$, how do i prove that $P\cap Q=1$? I was trying to use the inner automorphism to prove this, but i couldn't prove ...
2
votes
1answer
125 views

Question about Sylow Theorem and normalizer

I'm dealing with the following problem. Let $G$ be a finite group, $H$ and $K$ Sylow 3- 5- subgroups respectively of $G$. Suppose that 3 divides $|N(K)|$, show that 5 divides $|N(H)|$. I've ...
2
votes
1answer
84 views

Which of the following groups are subgroups?

I've written an answer for an exercise in Artin's algebra. Can someone please verify it? Which of the following groups are subgroups? (a) $GL_n(\mathbb{R}) \subset GL_n(\mathbb{C})$ (b) ...
2
votes
1answer
65 views

Characterisation of Galois Group with the action of $\sigma \in S_n$ on the roots

Let $f \in K[X]$ be irreducible and separable with roots $x_1,...,x_n$ in a splitting field $L$ of $f$ over $K$. We identify $\text{Gal}(L|K)$ with $\text{Gal}(L|K)\cong G\subset S_n$. How can I see ...
2
votes
2answers
79 views

Commutator subgroup having a complement

Suppose we have a finite group $G$ such that $[G,G]$ has a complement in $G$. Then what are some "good" things we can conclude from this situation? I know that "good" is not well-defined, but I am ...
2
votes
1answer
165 views

Question about $p$-Sylow subgroups of the quotient group

I have been working on the following problem. Let $G$ be a finite group, $N\trianglelefteq G$ and $p$ a prime, then $n_{p}(G/N)\leq n_{p}(G)$. I have beeen trying to solve it, but it seems I ...