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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3
votes
1answer
51 views

Prove the group is a direct product [closed]

Let $G$ be an abelian group of finite order $n = mk$ with gcd$(m,k) = 1$. For $r=m,k$, let $G(r) = \{g \in G: g^r = 1 \}$ . Prove that $G = G(m) \times G(k)$.
0
votes
0answers
28 views

Subgroups of finite reflection groups

I am trying to understand finite reflection groups. Given a connected finite reflection group generated by $m$ reflections and let $S$ be a set of simple roots. Let $I \subset S$ be a subset of the ...
1
vote
1answer
28 views

Finding cosets of a quotient group: List the cosets of $HN/N$

In the group $\Bbb Z_{24}$, let $H=\langle 4\rangle $ and $N=\langle 6\rangle $ 1) List the elements of $HN$. I found $HN=\{0,2,4,\cdots,22\}=\langle 2\rangle$ 2) List the elements of $H\cap N$. I ...
3
votes
0answers
28 views

How many symmetries does the Cauchy-Schwarz inequality have?

The symmetries of the Cauchy-Schwarz inequality $(x_1^2 + \cdots + x_n^2)(y_1^2 + \cdots + y_n^2) \ge (x_1y_1 + \cdots + x_ny_n)$, as a subgroup of the symmetric group on the $2n$ letters $x_1,\ldots, ...
1
vote
3answers
56 views

Is $H\cup K$ a group?

If $H$ and $K$ are subgroups of $G$ is $H\cup K$ also a subgroup of $G$? We have identity for sure(since it is in $H$ or $K$), associativity is absorbed. Thus we only need to see if inverses and ...
7
votes
1answer
94 views

What's the smallest non-$p$ group that isn't a semidirect product?

Although every non-simple group can be described as a group extension, it is well-known that not every non-simple group can be expressed as a semidirect product of smaller groups. For example, a ...
2
votes
0answers
51 views

Short exact sequence of groups

Let $1\to N\xrightarrow{\text{i}} G\to H\to 1$ be a short exact sequence of groups. Let $f:G\to N$ be a group homomorphism such that $fi: N\to N$ is an isomorphism. What are some "good" things we can ...
-1
votes
1answer
43 views

Soluble(solvable) and nilpotent groups

Defn 1.1. Let $\gamma _{0}(G)=G$, and $\gamma _{c}(G)=[\gamma _{c-1}(G),G]$ for $c\geq 1$. The lower central series of $G$ is a chain of subgroups of $G$: $$G=\gamma_0(G) \geq \gamma_1(G) \geq \cdots ...
3
votes
1answer
25 views

Hensel lifting when not a power of a prime

Say you have the equation $x^2 + x + 47 = 0$ and that you want to determine the solutions in $\mathbb{Z}/1715 \mathbb{Z}$. Note that $1715 = 7^3 \cdot 5$. Then, using Hensel's lemma, one can find the ...
1
vote
1answer
18 views

Extra-Special $p$ group and complement

Let $G$ be an extra special $p$ group of order $p^{2n+1}$, $n\geq 2$. Does $[G,G]$ necessarily have a complement in $G$? I dont think so, but I am not sure. Sorry this should be a very silly ...
0
votes
0answers
37 views

Number of groups of a certain order given: 1) finitely generated abelian, 2) subgroups, 3) not necessarily finitely generated

1) How many finite abelian groups are there of order $1000$? Well via the fundemental theorem of finitely generated abelian groups, we look at the factorisation for $1000$. $1000=2^3*5^3$ and there ...
1
vote
1answer
30 views

Subnormal series and indices

I'm trying to solve this problem: Let be $H, K$ subgroups of a finite group $G$. Suppose that exists one serie of subgroups such that $G=G_{0}\triangleright G_{1} \triangleright \ldots ...
2
votes
1answer
28 views

transitive subgroups of the symmetric group on $2(d+1)$ elements: can I always do at least $d$ permutations?

I have a set of $2(d+1)$ elements which are labelled as pairs $\{e_i, a_i\}_{i=1}^{d+1}$, transforming under some transitive subgroup of the symmetric group. This can be thought of as a regular ...
8
votes
1answer
60 views

Upper bounds for the number of intermediate subgroups

Assume that $G$ is a finite group, and $H\le G$ a subgroup of index $n>1$. What can we say about the number of distinct intermediate subgroups $K$, i.e. groups such that $H\subset K\subset G$? ...
0
votes
1answer
23 views

can i do this transformation with any finite group?

I have a finite alphabet $\{e_1, e_2, \cdot \cdot\cdot, e_N, a_1, a_2, \cdot\cdot\cdot, a_n \}$ where we pair $e_i$ and $a_i$ as ``opposites'' - like opposite vertexes on a regular $2N$ sided polygon ...
2
votes
2answers
207 views

Number of elements in a finite abelian groups

Is the following true? Let $G$ be a finite abelian group with a minimal generating set $S$. By minimal generating set I mean we cannot reduce the cardinality further. Let $S=\{a_1,a_2,\ldots ,a_k\}$ ...
2
votes
1answer
48 views

Can we see directly from the cocycle condition that 2-cocycles are symmetric?

Let $A$ be an abelian group and let $C$ be a cyclic group. All central extensions of $C$ by $A$ are abelian because in any such extension $$ 1\rightarrow A\rightarrow E\rightarrow C\rightarrow 1$$ ...
1
vote
1answer
44 views

What are the different subgroups of $A_4 \times \mathbb Z_2$?

Is there a place where I can find the subgroups of $A_4 \times \mathbb Z_2$ or is there a way I can list them completely? In particular, given the set of elements in terms of the generators, i.e., ...
0
votes
0answers
38 views

A probabilistic algorithm (something close to Discrete logarithm)

$p$ - is a prime number. $a\in Z_p^*$ is a creator of the group $Z_p ^*$. The definition of the Discrete logarithm of $c\in Z_p^*$ is: $$\log_a c=b\iff a^b=c \mod p $$ Assume we have an algorithm ...
2
votes
1answer
50 views

Subgroups of Abelian Group of order 1000

Suppose you have an abelian group of size 1,000. How many subgroups does it have? I know there are 9 such groups from $1,000 = 2^3 \times 5^3$ giving us 3 of order $2^3 \times$ 3 of order $5^3$ ...
1
vote
1answer
50 views

How to obtain real irreducible representation matrices for finite point groups?

I would like to generate the irreducible representation matrices in real (not complex) form for any finite point group, in order to use them in a projection operator. At least I require the diagonal ...
2
votes
1answer
26 views

List of two-sided wallpaper groups?

I'm interested in the symmetries of two-dimensional patterns that have two sides. In other words, what discrete groups can be formed from the three-dimensional Euclidean isometries which preserve a ...
4
votes
1answer
35 views

Geometric Representation of Quasidihedral Groups

I am going back through Dummit/Foote studying for a prelim and came across the 'quasidihedral' or 'semi-dihedral', group of order $2^n$, with presentation $\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs ...
1
vote
1answer
31 views

do these two groups isomorphic to each other?

consider two groups $T=<x,y |x^4=y^3=1,yxy=x>$ and $A=<x,y |x^6=1,x^3=y^2,xy=yx^{-1}>$, are these two groups isomorphic? I think this is not true,because $T$ don't have any 4 member ...
2
votes
1answer
23 views

2-Frobenius groups of order $2^{10}.3^5.5.11$

A group $G$ is called a 2-Frobenius group if $G=ABC$, where $A$ and $AB$ are normal subgroups of $G$, $AB$ is a Frobenius group with kernel $A$ and complement $B$ and $BC$ is a Frobenius group with ...
3
votes
1answer
79 views

If $C_G(H)=N_G(H)$ for all abelian subgroups, prove that $G$ is abelian

Let $G$ be a finite group such that for all abelian subgroups $H$ of $G$, $$C_G(H)=N_G(H).$$ Prove that $G$ is abelian. ($C_G(H)$ is the centralizer, $N_G(H)$ is the normalizer of $H$ in $G$) my ...
1
vote
2answers
35 views

Uniqueness of a subgroup of a given order

Let $G$ be a cyclic group with order $n$. Prove that for every divisor $d$ of $n$ there is a unique subgroup with order $d$. For the existence, let $x$ be a generator of $G$. It is easy to check ...
0
votes
1answer
74 views

Why every finite non-abelian simple group of order $n$ has a proper subgroup of index at most $kn^{\frac{3}{7}}$? [closed]

Can anyone please clarify me on why every finite non-abelian simple group of order $n$ has a proper subgroup of index at most $kn^{\frac{3}{7}}$ for some positive constant $k$? Thanks
0
votes
0answers
36 views

Size of conjugacy classes of alternating group $A_{22}$

Let $p,q\in\{13,17,19\}$ and $G=A_{22}$, is it true that for every $x\in G$ we have $(|x^G|,pq)\neq 1$? why?
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vote
0answers
34 views

Sum of squares in finite field cannot be congruent to $0$? [closed]

Consider a finite field $GF(p)$, where p is a prime integer and $p\equiv 3 (mod 4)$. Consider two elements $a,b\in GF (p)$. How to prove $a^2+b^2\neq 0 (mod p)$.
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0answers
22 views

Finitely generated subgroups of $\mathbb Q$ [duplicate]

Is it true that any finitely generated subgroup of $\mathbb Q$ is infinite cyclic? My try: if $I=<a_1/b_1,...,a_n/b_n>$ is a f.g. $\mathbb Z$-submodule of $\mathbb Q$ then all $a_i$'s lie in ...
0
votes
1answer
30 views

Action on finite non-abelian group

Let $G$ be a finite, non-abelian group. Show that if $Aut(G)$ acts on $G$ by $\sigma.g=\sigma(g)$ for each $\sigma \in Aut(G)$, $g \in G$, then there exist at least three orbits. I think I could ...
0
votes
0answers
15 views

Finite $2$-groups of order $>32$ and nilpotency class $2$

I only know $2$-groups of nilpotency class $2$ and order less than or equal to $32$, and wondering if there are finite $2$-groups of order $>32$ and nilpotency class $2$? Your suggestions are ...
0
votes
3answers
40 views

Given a group $G$ and subgroup $H<G$ and $g \in G$ such that $gHg^{-1} \subset H$ prove

My problem is the following - Given a group $G$ and subgroup $H<G$ and $g \in G$ such that $gHg^{-1}\subset H$ prove that if H is a finite group $|H|< \infty$ then $gHg^{-1} = H$ what I tried ...
1
vote
1answer
62 views

Characterizing the cosets of a cycle of a finite abelian group with a linear combination of floor functions

Prelude Cconsider the finite abelian group $\mathcal G = \prod_{i=1}^A \mathbb Z_{a_i}$ and let $\mathbf s \in \mathcal G$. Let $\mathcal H = \operatorname{grp}({\mathbf s})$ be the subgroup of ...
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vote
3answers
49 views

Showing that $3$ is a generator of the group $(\mathbb{Z}/31\mathbb{Z})^{\times}$

Show that $3$ is a generator of the group $(\mathbb{Z}/31\mathbb{Z})^{\times}$. I have done the following: The order of the group is $30=2 \cdot 3 \cdot 5$. $3$ is a generator if $3^2, 3^3, ...
2
votes
1answer
75 views

Positive Elements of $\mathbb{C}G$: as functionals versus as elements of the C*-algebra

I might have thought about this problem a little longer but am quite confused so said I would put this question to the good people here... Consider a finite group $G$ or rather the algebra of ...
0
votes
2answers
90 views

How to prove that a given group is isomorphic to Sym(4)?

Given a specific group with 24 elements, I want to prove that it is isomorphic to Sym(4). To begin with, I calculate the orders of my group's elements and they come out as in the order statistics for ...
2
votes
2answers
41 views

Let $G$ be a finite group, $ord(G)=p^2$ ($p$ is a prime) prove that there is a subgroup of order $p$ in $G$

Let $G$ be a finite group, $ord(G)=p^2$ ($p$ is a prime) prove that there is a subgroup of order $p$ in $G$ I thout about Sylow theorem but it didn't helped me.
0
votes
1answer
50 views

Is there any group $G$ with $ord(G)=20$ so that $\varphi:G\rightarrow \mathbb{Z}_{15}$ is epimorphism?

Is there any group $G$ with $ord(G)=20$ so that $\varphi:G\rightarrow \mathbb{Z}_{15}$ is epimorphism? I thout about $\mathbb{Z}_{20}$ to map $\varphi:\mathbb{Z}_{20}\rightarrow\mathbb{Z}_{15}$,such ...
0
votes
1answer
34 views

Is there any group $G$ with $ord(G)=20$ so that $\varphi:G\rightarrow \mathbb{Z}_{10}$ is epimorphism?

Is there any group $G$ with $ord(G)=20$ so that $\varphi:G\rightarrow \mathbb{Z}_{10}$ is epimorphism? I thout about $\mathbb{2\cdot Z}_{40}$, is it right?
2
votes
2answers
80 views

find the number of elements of order 3 in an abelian group of order 120

Let order of G=120. Then the number of sylow 3 grs are (1+3k)=p.where p divides 8. So k=0 or 1. Which one i take?
2
votes
1answer
78 views

Find the smallest positive integer n such that there are exactly four non-isomorphic abelian groups of order n

What is the smallest positive integer n such that there are exactly four non-isomorphic abelian groups of order n? This is a question in Joseph A.Gallian's book, and the answer is n=36 and the ...
1
vote
0answers
19 views

Cycle Structure of a Permutation Based on the Binary Representation

Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number as follows. Given a non-negative integer $k$, let $s(k)=\frac{b+1}{2}$, where $b=\max\limits_c\big(c2^k\le n, ...
4
votes
1answer
57 views

Can we always construct a “$p$th root” of a $p$-element in a finite group?

Let $p$ be a prime, $G$ a finite group, and $g\in G$ a $p$-element. Can one always embed $G$ in a finite group $H$ that contains a $p$-element $h$ such that $h^p=g$?
1
vote
1answer
29 views

Obtaining a presentation of the dihedral group from a semidirect product

I am working on classifying groups of order 44. I have shown that $G\cong P_{11} \rtimes_{\varphi} P_{2}$, where $P_p$ are Sylow p-subgroups and $\varphi:P_{2} \to $Aut$(P_{11})$ is a group ...
0
votes
1answer
47 views

if $G$ is finite group then polycyclic group is equivalent to super solvable group?

I don't know why this is true? can you help me: if $G$ is finite group then polycyclic group is equivalent to super solvable group Definitions- Polycyclic group $G$ is a polycyclic if has a ...
1
vote
1answer
51 views

Colouring a tetrahedron

How would I write down the elements sr and $sr^2$ of G as a product of disjoint cycles? If I am looking for the orbits of this action, do I have 4 orbits $\{1\:2\:3\}, \{p12\:p23 ...
3
votes
2answers
53 views

$G$ a finite group, $H$ a subgroup of index $2$ in $G$. If $K$ a subgroup of $G$ of odd order then $K$ contained in H.

Let $G$ be a finite group and $H$ a subgroup of $G$ such that $|G:H|=2$. Suppose $K$ a subgroup of $G$ of odd order. Show $K$ is contained in $H$. I'm stuck. Need a hint.
5
votes
2answers
63 views

Classification of all finite elementary $p$-groups.

Let $G$ be a finite group. For a prime number $p$, let us call $G$ an elementary $p$-group iff $\exp G=p$. I know that all elementary $2$-groups are abelian, and I also know the construction of ...