Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

learn more… | top users | synonyms

7
votes
2answers
278 views

Facts about Abelian Groups and group order.

I look for some theorems which tell us about the relation between the property of being abelian for groups and the order of the group. I think these theorems are provided in a second course of group ...
56
votes
8answers
7k views

Are there real world applications of finite group theory?

I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics. (Sufficiently applied mathematics such as ...
5
votes
2answers
218 views

How to count the conjugates of an exotic $S_5$?

It can be read off the The Elliott configuration - a $5$-coloring of $K6$ - that $S_6$ has an exotic $S_5$ subgroup (it's not a point stabilizer) which I will call $X_5 = \langle (1\;3\;6\;4\;5), ...
11
votes
1answer
853 views

Simple groups of order 168

How would I prove that there is at most one simple group of order 168? I've already seen that $GL_3(2)$ and $PSL_2(7)$ are simple groups of order 168, and I have seen direct proofs that they are ...
2
votes
3answers
145 views

If $|G|=65$ then $G$ contains an element of order $5$

If $|G|=65$ then $G$ contains an element of order $5$. Proof: If $G$ is cyclic then there is some $x \in G$ where $|x| = 65$ so $|x^{13}|=5$. Assume $G$ is not cyclic and that no elements in $G$ ...
8
votes
3answers
823 views

Prove that a group with exactly two proper nontrivial subgroups is isomorphic to $\mathbb{Z}_{pq}$ or $\mathbb{Z}_{p^3}$.

Suppose $G$ is a group and has exactly two nontrivial proper subgroups. Prove that $G$ is cyclic and $|G|=pq$ where p,q are distinct primes or $G$ is cyclic and $|G|=p^3$ where $p$ is a prime. Usually ...
1
vote
1answer
48 views

If we have $f$ is one-to-one, why can we conclude that $n\mathbb{Z}_m=\mathbb{Z}_m? $

Suppose $m,n \in \mathbb{Z},m,n\geq1.$ Define a map $$f:\mathbb{Z}_m \rightarrow \mathbb{Z}_m$$ where $[x] \rightarrow [nx]$ If we have $f$ is one-to-one, why can we conclude that ...
3
votes
3answers
610 views

Showing that a transitive abelian permutation group is necessarily regular

I am trying to show that a transitive, abelian permutation group acting on a set $X$ is necessarily regular, given this hint: 'Given $g \in G$, consider the set $X^g:=\{x \in X\,|\,gx=x\}$. Show that ...
3
votes
3answers
107 views

$D_6$ as permutation group

I am trying to solve some exercises for a course in representation theory. We are studying finite groups and I have an exercise about the dihedral groups $D_n = ...
2
votes
1answer
225 views

Completing a Cayley table with few given spaces

\begin{array}{ccc} * & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} \\ \textbf{1} & 1 & & & & & \\ \textbf{2} & ...
4
votes
1answer
320 views

I don't understand symmetries of the Fano plane

Hello I got a picture of the Fano plane, but there are 5 points on every line. Why aren't there 3? And I cannot see what it's symmetry group is. I have been told it's $PSL_2(7)$ but that doesn't ...
1
vote
1answer
108 views

The set that a group can act on it as a permutation group

What can we say about the set that a group can act on it as a permutation group? My question is about the structure of elements of a group G when acts on an arbitrary set Ω as a permutation group. For ...
3
votes
2answers
106 views

$\pi$-radical of group

Let $G$ be abelian and periodic. Let $\mathbb{P}$ be the set of prime numbers, $\pi \subseteq \Bbb P$ and $\pi ^{\prime }=\Bbb P\setminus\pi $. Let $O_{\pi }\left( G\right) =\left\langle ...
3
votes
1answer
105 views

Conjugate of an abelian maximal subgroup is maximal

Perhaps this is a trivial question. Let $G$ be a finite group, and $M$ be a maximal (proper) subgroup of $G$. Suppose also that $M$ is abelian. How could I prove that if $x\in G$, then $xMx^{-1}$ is a ...
2
votes
4answers
876 views

Finite Group Proving finite order of elements and Subgroup Question

The question is as follows Let G be a finite group. (i) Prove that every element of G has finite order. For this want to use the idea that if G is finite then for a in G, $a^{n}$ = $e$ for some n ...
7
votes
3answers
169 views

Problem involving permutation matrices from Michael Artin's book.

Let $p$ be the permutation $(3 4 2 1)$ of the four indices. The permutation matrix associated with it is $$ P = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 ...
2
votes
1answer
100 views

Group Theory. Transitivity and Normal subgroups.

I would like to show that, If $G$ acts transitively on a set $X$, and $K$ is regular normal subgroup of $G$. Then $G = K \operatorname{Stab}(a)$. ($K \operatorname{Stab}(a)$. w.r.t G) for any $a \in ...
2
votes
1answer
82 views

Group extension

I am getting blurred about group extensions. Let $A,B,C$ be groups. If $G=(A{:}B).C$ and $A$ is characteristic in $G$, then $G=A.(B.C)$. But is it also true that $G=A{:}(B.C)$ ? If $G=A.(B{:}C)$, ...
17
votes
3answers
729 views

Is there a geometric realization of Quaternion group?

Is there a geometric realization of the Quaternion group: $$Q = \langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle$$ I dont think it can be realized as the symmetries/rotations of a 3D shape so could ...
17
votes
1answer
196 views

References on the theory of $2$-groups.

Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to ...
5
votes
2answers
504 views

Generators of Symmetric and Alternating Group

Consider the symmetric and alternating groups $S_n$ and $A_n$ ($n>2$). 1. Does arbitrary $2$-cycle and an arbitrary $n$-cycle in $S_n$ generates $S_n$? 2. If $n$ is odd, does an arbitrary ...
11
votes
4answers
2k views

There exists only two groups of order $p^2$ up to isomoprhism.

I just proved that any finite group of order $p^2$ for $p$ a prime is abelian. The author now asks to show that there are only two such groups up to isomorphism. The first group I can think of is ...
1
vote
1answer
53 views

Understand group extensions

I will try to explain what I know so far (not certain is it right though): A group extension is denoted by $G = N.Q$ or $N:Q$ (the second case is a semidirect product) I think this just means $G/N ...
1
vote
1answer
51 views

Subgroups of minimal non-abelian 2-groups

I want to determine all maximal subgroups of the following type of minimal non-abelian 2-group: $\langle a, b, c : a^{2^{s}} = b^2 = c^{2} = 1, [a,c] = [b,c] = 1, [a, b]=c\rangle$ of order $2^{s+2}$; ...
3
votes
2answers
1k views

How many non isomorphic groups of order 30 are there?

Let $|G|=30$. I have prove that there is the only subgroup of order $15$, which I'll denote $H$. Now I do know how to classify the group. After thinking, I made the following steps. 1) Possible ...
6
votes
2answers
294 views

If $P_1 , P_2 $ are two $p$-sylow subgroups, prove that $ P_1 \bigcap $ $P_2$ = $ { 1 } $

If $P_1 , P_2 $ are two sylow $p$-subgroups of the group $G$ prove that: $ P_1 \bigcap $ $P_2$ = $ { 1 } $ I tried to prove it by induction as follows: proved it when $P_1 , P_2$ have the ...
4
votes
1answer
199 views

On Group of order $30$ and $60$.

In this question on yahoo answers , the answer says , "with $t = 6$, then there are 6 * (5 - 1) = 24 elements of order $5$ " my question is , how did " 6 * ( 5 - 1 ) " come from ? Which ...
1
vote
2answers
132 views

Degree of a permutation group

What can we say about the set that a group can act on it as a permutation group? we know that this set is not unique. For example the alternative group $A_4$ acts on the sets of sizes 4 and 6.
0
votes
0answers
93 views

Propery of generated subgroup by conjugacy class

I want know some fact about generated subgroup by conjugacy class of a element in arbitrary group G.please help me.
0
votes
4answers
89 views

In $Q_8$ why $C_G(i)=C_G(-i)$

$Q_8$ is the Quaternion group. In $Q_8$ why $C_G(i)=C_G(-i)$?
2
votes
2answers
113 views

Existence of a subgroup

Let $G$ be a finite abelian group and $H$ a subgroup of $G$. Then there exists a subgroup $L$ of $G$ such that $L≃G/H$.
0
votes
1answer
161 views

Why must a finite symmetry group be discrete?

I'm having trouble justifying why a finite symmetry group is discrete. Can someone help?
2
votes
1answer
80 views

Are these maps group homomorphisms?

Let $G$ be a finite group, $M$ a trivial $G$-module and let $f:G \times G \to M$ be a 2-cocycle. Question: Are the following maps $f_1,f_2$ group homomorphisms ? $$f_1: G \to M,\; g \mapsto ...
1
vote
1answer
287 views

On the Group of order $pq$ where $p , q $ are primes .

Let $G$ be a group, $\lvert G \rvert = pq$ where $p$, $q$ are primes, $q$ is bigger than $p$. Let $P$ be a Sylow $p$-subgroup and $Q$ be a Sylow $q$-subgroup and let $n_p$= the number of Sylow ...
7
votes
1answer
297 views

Normalizer of $S_n$ in $GL_n(K)$

In the exercises on direct product of groups of Dummit & Foote, I proved that the symmetric group $S_n$ is isomorphic to a subgroup of $GL_n(K)$, called the permutation matrices with one 1 in each ...
3
votes
2answers
92 views

Existence of a certain finite $2-$group

Is there a finite non-abelian $2-$group $P$ of order $2^n; n\geq 4$ such that $\frac{P}{\langle x \rangle}\cong \Bbb{Z}_2\times\Bbb{Z}_{2^{n-2}}$ for $x\in Z(P)$ and $\lvert x\rvert =2$? ($\lvert x ...
1
vote
1answer
57 views

For which values of $n \ge 2$ is $H_n = \{\alpha \in S_n:|\alpha| \text{ is odd}\}$ a subgroup of $S_n$?

Let $H_n = \{\alpha \in S_n:|\alpha| \text{ is odd}\}$. For which values of $n \ge 2$ is $H_n$ a subgroup of $S_n$? Ok so I figure that since the order is odd, then $\alpha$ can be written as a ...
1
vote
2answers
156 views

How to prove $|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})]$?

Let $G$ be a finite group and $H \leq G$, $K \leq G$. For an element $g \in G$, prove $$|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})].$$ I am not sure where to start proving this. A hint would be ...
2
votes
1answer
83 views

Details about “fingerprinting” algorithms for groups?

where can I find details about "Fingerprinting" algorithms (to test whether two groups are non-isomorphic) "‘Fingerprinting’: For every group $G_1,…, G_r$ evaluate various isomorphism-invariant ...
1
vote
1answer
63 views

A finite simple group with $\pi (G)\subseteq \pi (p^{2}-1)$

Let $\pi (k)$ be the set of prime divisors of $k$ and let $\pi (G)=\pi (|G|)$. Let $G$ be finite simple group with $\pi (G)\subseteq \pi (p^{2}-1)$, where $p$ is prime. I would like to know is there ...
1
vote
1answer
94 views

Relation between a representative of a conjugacy class and corresponding irreducible character value

Is there a relation between the representative order of a conjugacy class and the corresponding irreducible character value? Thanks in advance.
4
votes
2answers
990 views

Finding the number of elements of order two in the symmetric group $S_4$

Find the number of elements of order two in the symmetric group $S_4$ of all permutations of the four symbols {$1,2,3,4$}. the order two elements are two cycles.number of $2$ cycles are $6$.but the ...
5
votes
1answer
91 views

what are the algorithms available to get all the elements of a finitely presented group?

Is coset enumeration the best way to get all the elements of a finitely presented group? if not what is the best algorithm to do this ? what difficulties could be caused by this process ( I mean ...
2
votes
1answer
118 views

Questions about products of $p$-cycles.

Let $p$ be a prime and let $n$ be an integer such that $n \le p$. a) Let $\alpha$ be a p-cycle in $S_n$. Prove that $\alpha^j$ is a p-cycle for $1 \le j \le p − 1$. b) Assume that $2p \le ...
1
vote
1answer
94 views

About direct products of Groups

Let $G$ be a finite soluble group. Let $T$, $S\leq G$ such that $T$ and $S$ are nilpotent ($T=T_{p_1}\times \cdots \times T_{p_n}$) and $T^{*}= T_{p_1}^{g_1}\times\cdots\times T_{p_n}^{g_n}\leq S$, ...
3
votes
2answers
134 views

Sequence and series problems with slick group theoretic solutions.

So I'm trying to get this unmotivated student in my class interested in learning group theory. I've recently ascertained that he likes analysis a bit better than algebra, and that sequences and ...
31
votes
1answer
504 views

Sudokus as composition tables of finite groups

If $G$ is a finite group then the composition table of $G$ is a latin square (ie, each row and column contains each group element exactly once). Assume now that $|G| = n^2$ for some natural number ...
1
vote
0answers
83 views

Efficiency of Group-Theoretic Algorithms in MAGMA

Given a finite permutation group $G$ and an element $a\in G$ with conjugacy class $X$, I am interested in determining when for a given element $x\in X$ the subgroup $<a,x>$ generated by $a$ and ...
4
votes
3answers
147 views

Determine if $G$ is a group under the $\,\gcd\,$ operation

Let $G = [1,2,3,4,6,12].\;$ Let $\,a*b = \gcd(a,b), a,b \in G.\;$ Determine whether $G$ is a group. I have found that for any two elements in $G$, commutativity holds, but the inverses are not ...
1
vote
5answers
136 views

Permutation question. $qpq^{-1}$, $q,p,r,s \in S_{8}$.

Let $p,r,s,q \in S_{8}$ be the permutation given by the following products of cycles: $$p=(1,4,3,8,2)(1,2)(1,5)$$ $$q=(1,2,3)(4,5,6,8)$$ $$r=(1,2,3,8,7,4,3)(5,6)$$ $$s=(1,3,4)(2,3,5,7)(1,8,4,6)$$ ...