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.

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-2
votes
0answers
16 views

Expressing $z\in G$ as $z=gh^2$

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
0answers
21 views

Does the order of a finite group divide the product of degrees of a system of parameters of the invariant algebra?

Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ ...
6
votes
3answers
48 views

$H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\} , \forall g \in G \setminus H$ , then $|\cup gHg^{-1}|>\dfrac 12 |G|+1$

Let $H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\}$ for all $g \in G \setminus H$. Then is it true that $$|\cup_{g \in G \setminus H}gHg^{-1}|>\dfrac 12 |G|+1$$ If ...
0
votes
1answer
18 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 ...
1
vote
0answers
30 views
+50

If $M < M < G$ with certain conditions and a special subgroup $U < G$, then we can choose $|G / M| = p$ and $p \nmid |M|$.

Let $G$ be a finite group and $U \le G$ a subgroup of odd order. Assume that $N_G(U) = TU$ with $T = \langle t \rangle$ for some involution $t \notin U$. Also assume $U^g \ne U$ implies $U^g \cap U = ...
0
votes
1answer
25 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 ...
1
vote
2answers
12 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 $12 \mod 4 = 0$ ...
1
vote
0answers
22 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 ...
1
vote
3answers
48 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 ...
3
votes
1answer
43 views

Is it enough to determine if two finite groups are isomorphic if we can map the generators?

Heading: A General Inquiry about Finite Isomorphic Groups and Their Generators If I am given two finite groups whose generators I know, is it enough to show that by mapping the generators ...
15
votes
1answer
4k 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 $\lambda_x (g) = xg$. ...
1
vote
1answer
16 views

Semidirect product when the action factors

Suppose I am forming a semidirect product of finite groups $G \rtimes_\theta A$. Here, $\theta : A \to \mathrm{Aut}(G)$ is a homomorphism. Now, suppose I notice that the homomorphism $\theta$ factors ...
6
votes
2answers
105 views
+300

How are simple groups the building blocks?

I know a bit about simple groups. A finite Abelian group is a (direct) product of finite cyclic groups. The simple finite Abelian groups are exactly $\mathbb{Z}_p$ for $p$ a prime. And so, I ...
1
vote
0answers
37 views

Why is $M_{21}\cong PSL_3(\mathbb{F}_4)$?

I have recently been reading about the Large Mathieu Groups in "The Finite Simple Groups" by Robert Wilson and I have not come across any explanation of this isomorphism as of yet. In this context, ...
3
votes
1answer
47 views

How can I check whether the group $[16,13]$ in GAP with $3$ generators can be generated by $2$ elements?

The group $[16,13]$ in GAP has structure $(C4\times C2):C2$ and is generated by the permutations $(1234)(5678)$ , $(15)(26)(37)(48)$ and $(57)(68)$ . The group $[16,3]$ in contrast with the same ...
3
votes
1answer
50 views

Which non-abelian finite groups have the property that every subgroup is normal?

If $G$ is an abelian group, every subgroup $H$ of $G$ is normal. I searched for non-abelian finite groups $G$ , such that every subgroup is normal and GAP showed only the groups $G'\times Q_8$ , ...
-1
votes
2answers
73 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?
1
vote
0answers
6 views

Alternative construction of the twisted group $^2 E_6 (q)$.

I am looking for the alternative construction of the twisted finite simple group $^2 E_6 (q)$ possibly avoiding the Lie theory. Especially I am interested in calculating its order. Maybe some of you ...
0
votes
0answers
36 views

How can I prove that $G$ is not a simple group. [duplicate]

I only have that $|G|=4n+2$ for $n \in \mathbb{N}$. And I have to prove that G is not a simple group.
1
vote
1answer
15 views

Why is the Young symmetrizer non-zero?

Suppose $\lambda$ is a partition of the natural number $n$ and $T$ is a standard Young Tableaux of shape $\lambda$. Let $$P_{\lambda}:=\lbrace g\in S_n:g\text{ preserves the rows of }T\rbrace$$ and ...
1
vote
2answers
34 views

$G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$?

Let $G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$ ? ( I know that any subgroup of index smallest prime dividing ...
0
votes
0answers
31 views

Quotient of $S_4$ by a normal subgroup

Is this true? > Quotient of $S_4$ by a normal subgroup of order 4 is Abelian ? As the group is of order 4!/4=6, so it is either $S_3$ or $\mathbb Z_6$. how to decide? please help.
0
votes
0answers
20 views

Image of Hall subgroup is a Hall subgroup

Let $\pi$ be any set of prime numbers. $H \subset G$ is a Hall $\pi$-subgroup if no pime dividing the index $|G:H|$ lies in $\pi$ and every prime divisor of $|H|$ lies in $\pi$. Let $\varphi: G ...
2
votes
1answer
45 views

Is the definition given by the GAP-manual equivalent to the one given in the site? [on hold]

Here http://groupprops.subwiki.org/wiki/Normalizer_of_a_subset_of_a_group the normalizer of a subset of a group is defined. GAP gives the following description of the Function IsNormal : 39.3.6 ...
1
vote
1answer
37 views

If $G = G_{\alpha} \rtimes R$ and $R$ is non-abelian of order $27$ and exponent $3$, then $G_{\alpha} \cong C_2\times C_2, D_3$ or $D_4$

Suppose that $G$ is a transitive permutation group and let $R$ be a regular normal subgroup which is isomorphic to the non-abelian group of order $27$ and exponent $3$. Then $G = R \rtimes ...
2
votes
0answers
18 views

Factorization of permutations into two factors with fixed number of cycles, plus a placement condition

I have asked this question in MathOverflow, but it received no answers, so I am posting it here. In my recent work I have been led to consider the following type of permutation factorizations. Let ...
1
vote
1answer
21 views

Normalizer probtlem for finite nilpotent groups

Lemma- Suppose $P$ is a $p$-group contained in $G$ and $u\in N_U(G)$ where $U=U(\Bbb{Z}G)$. Then there exist $y\in G$ such that $u^{-1}gu=y^{-1}gy\ \forall\ g\in P$. We use this lemma to prove ...
0
votes
1answer
38 views

The trivial homomorphism [on hold]

Please help me I need to prove that for any positive integer n, the only homomorphism between the cyclic group of order n and the complex number under addition is the trivial homomorphism
2
votes
2answers
34 views

$P$ is a Sylow subgroup of $G$ then $P$ is normal in $G$.

Let $G$ be a finite group with subgroups $H$ and $P$ and if $H$ is normal in $G$ and $P$ is normal in $H$ and $P$ is a Sylow subgroup of $G$ then $P$ is normal in $G$. Is the statement true, I heard ...
2
votes
1answer
35 views

If $[G:H\cap K]= [G:H][G:K]$ then $G=HK$.

Let $G$ a finite group and $H,K$ subgroups of $G$. Show that if $[G:H\cap K]= [G:H][G:K]$ then $G=HK$. I proved that $G=HK$ implies $[G:H\cap K]= [G:H][G:K]$ but not the other direction. Thanks for ...
1
vote
0answers
18 views

Multiplication table of $(S_1 \wr D_2) \times (S_1 \wr D_3)$

I am trying to compute the multiplication table of $(S_1 \wr D_2) \times (S_1 \wr D_3)$. My effort: I understand that $S_1$ is the trivial group consisting only the unit element i.e. $e$. $e$ will ...
0
votes
0answers
13 views

Computing character for the partition $(2, 2, 1, 1)$ using Murnaghan–Nakayama rule

I am trying to understand an example from Murnaghan–Nakayama rule as it is described in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. The group ...
0
votes
0answers
18 views

Understanding the Murnaghan–Nakayama rule

I am trying to understand the Murnaghan–Nakayama rule as it is described in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. Here is the ...
0
votes
1answer
35 views

Permutation representations of symmetric group of small order

Thanks for any comments or help. What is the list of faithful permutation representations of $S_k$ of degree at most $n=2k$ for $k=3,4,5,6$? Is it possible to find its centralizer in each case?
0
votes
0answers
13 views

Murnaghan–Nakayama rule for order 2 subgroup of symmetric group

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. In section 5, a scenario is presented as follows. Here, $G$ is the disjiont ...
3
votes
0answers
47 views

What dihedral subgroups occur in the affine general linear group $AGL(2,3)$

I am interested in the subgroup structure of the affine general linear group $AGL(2,3)$, in particular I want to know if they could have dihedral subgroups other then $D_3$ and $D_4$, i.e. the ones of ...
3
votes
1answer
42 views
0
votes
1answer
38 views

Is the dihedral group Dn linearly primitive for n>2?

Let $D_n$ be the Dihedral group (of order $2n$). For $p>2$ a prime number, $\mathbb{Z}/2$ is a core-free maximal subgroup of $D_p$, then $D_p$ is a primitive permutation group, and so linearly ...
5
votes
2answers
156 views

Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$. My thought. Since ...
4
votes
1answer
52 views

Automorphisms of a group and subgroup

Is there a finite $p$-group $G$ such that $G$ has less number of automorphisms than some subgroup: $$|\mbox{Aut}(G)|<|\mbox{Aut}(H)| \mbox{ for some }H\leq G.$$ If there is such a group, then can ...
2
votes
1answer
55 views

Can the possible groups by determined by hand?

Suppose, $G$ is a group of order $12$ containing an element $a$ with order $4$. Can I show the following facts by hand ? The group is either cyclic or isomorphic to the group $C3:C4$ $a^2$ is the ...
0
votes
2answers
31 views

Is there a cyclic group such that is isomorphic to Z∗16? [closed]

How do I find an additive group modulo $n$ $Z_n$ which is isomorphic to $Z^*_{16}$ (group with multiplication modulo 16)?
1
vote
1answer
25 views

An inequality for the minimal number of generators of a finite group II

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating exactly the left regular representation (with ...
7
votes
0answers
214 views

An inequality for the minimal number of generators of a finite group

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular ...
13
votes
2answers
1k views

$A_n$ is the only subgroup of $S_n$ of index $2$.

How to prove that the only subgroup of the symmetric group $S_n$ of order $n!/2$ is $A_n$? Why isn't there other possibility? Thanks :)
0
votes
1answer
27 views

Problem 4 of Section G of Chapter 13 of Pinter's Book of Abstract Algebra

First, some background info/context: Let $G$ be any group of order $10$. Then, by Cauchy's Theorem, there are elements $a, b \in G$ such that $\text{ord}(a)=2$ and $\text{ord}(b)=5$. Since ...
4
votes
4answers
135 views

Group conjecture

Conjecture: Given a finite group $G$ and a subset $A\subset G$. Then $\{A,A^2,A^3,\dots\}$ is a group iff $\forall n\in \mathbb N: |A^n|=|A^{n+1}|$. Given that the composition between the ...
1
vote
0answers
35 views

Can the number of order-sequences of a group of order $n$ be determined efficiently?

Let $n\ge 1$ and $G$ a group of order $n$. Determine the order of every element of $G$ and list the orders increasing. Can the number of possible order-sequences be determined efficiently ? For ...
0
votes
0answers
57 views

Can a group of order $385$ be non abelian? [closed]

I was asked if all groups of order $385$ are abelian or not I know its $7$ sylow group is central but cannot really proceed. Can someone please help?
15
votes
1answer
417 views

Values attained by $|G/Z(G)|$?

So I was working through some problems in a book on $p$-groups and noticed that $p$-groups have some really nice properties. So I started computing what the values of $|G/Z(G)|$ for $p$-groups. I ...