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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4
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1answer
27 views

Unique normal subgroups of every possible order

Can one characterize groups $G$ for which there is a unique normal subgroup of order $d$ for every divisor of the order of $G$? For example must they be solvable?
3
votes
4answers
97 views

If a group contains a subgroup with the order of each of its divisors, is it abelian?

Let $G$ be a group that has a subgroup of size $d$ for every $d$ that divides $|G|$. Must $G$ be abelian? It can be shown using complete induction that the converse of the above statement is true, ...
1
vote
1answer
27 views

Characterization of solvable groups in terms of subgroups of certain orders?

In this question, the OP mentions the following result: a finite group $G$ is solvable if and only if $$\text{for all $n$ dividing $|G|$ such that $\gcd(\frac{|G|}{n},n)=1$, $G$ has a subroup order ...
0
votes
1answer
45 views

Find a subgroup of $S_4$ that is isomorphic to V, the Klein group.

So I know that the Klein group is the group with 4 elements that is not cyclic but I'm stuck from there onwards?
0
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1answer
40 views

The number of subgroups conjugate to a given subgroup of a finite group

Let $H$ and $K$ are subgroups of $G$ conjugate to each other. $A$ is defined as $$A = \{a \in G \mid aHa^{-1} = H \}$$ for all $a\in G$. Prove that $A$ is a subgroup of $G$ and prove that if $G$ ...
0
votes
1answer
41 views

2 question on solvable group's property

A theorem says a finite group G is solvable if and only if for every divisor n of $\vert G\vert$ such that (n,$\frac{\vert G\vert}{n})=1$,G has a subgroup of order n. Does this imply that G must ...
1
vote
0answers
42 views

how is the jordan-holder theorem used in conjunction with short exact sequences to construct groups of certain order?

I am an undergraduate and have been asked to explain how simple groups can be used to construct groups of finite order. I started with reading about the extension problem in group theory and from ...
0
votes
1answer
75 views

How to prove that every nonabelian group of odd order is not simple?

How to prove or disprove that every nonabelian group of odd order is not simple? I have no ideas concerning that.
3
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0answers
38 views

Groups of order $n^2$ with no subgroup of order $n$ [duplicate]

Is it possible to classify those groups whose order is $n^2$ for some natural number $n$ but which do not have any subgroups of order $n$? To be a bit more specific (in case a full classification is ...
9
votes
2answers
123 views

Groups of order $n^2$ that have no subgroup of order $n$

For which $n$ is there a group of order $n^2$ without a subgroup of order $n$. Such groups can not be nilpotent. This question is related to Sudokus as composition tables of finite groups.
0
votes
0answers
30 views

A proposition to check two isomorphic nonabelian finite simple groups [on hold]

Let $S$ and $T$ be two nonabelian finite simple groups and $G=S \times T$. How do we prove the following proposition: S and T are isomorphic if and only if G has a maximal proper subgroup which ...
0
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1answer
50 views

N is a normal subgroup of G if $aNa^{-1} \subset N $ for all $a ∈ G$. Prove that in that case, $aNa^{-1} = N $.

I said if N is a normal subgroup of G when $aNa^{-1} \subset N $ aN = Na as N is a normal subgroup of $G$. Therefore $aNa^{-1} = Naa^{-1} $ and $aNa^{-1} = N $. I would like to go with this proof ...
2
votes
1answer
64 views

Structure of a group, $G$, of order $pq$ where $p, q$ are prime.

There is a proposition in Beachy and Blair's Abstract Algebra that I don't entirely follow. The proposition is the following: Let $G$ be a group of order $pq$, where $p > q$ are primes. a) If ...
1
vote
0answers
35 views

Counting apartments in spherical buildings

Is there a formula for the number of apartments in a finite, spherical building? To be specific, is there a formula that depends on the associated Coxeter group and the thickness of the building? ...
3
votes
1answer
50 views

An elementary question regarding a multiplicative character over finite fields

Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I ...
0
votes
4answers
74 views

If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$

I've independently come up with a question (I know it's been asked before, but I can't find the question online) involving the external direct product, non-cyclic groups and isomorphisms. So, is the ...
2
votes
1answer
43 views

Relation of order of a permutation with its sign

Let $G$ be a group with order $2m$ where $m$ is odd. Consider the left action $\lambda_g:G\to G$. It appears that if $g$ has odd order iff $\lambda_g$ has odd order iff $\lambda_g$ is an even ...
0
votes
1answer
33 views

pemutation representation that confuses me a lot recently

For any group G, define group action on a set A. There will be a permutation representation of that group action. I am kind of confused why the permutation representation can be used to reflect the ...
3
votes
0answers
29 views

Show that Icosahedral group does not have normal subgroup of order 5

Here is my work so far: Let $G$ denote the group of orientation-preserving isometries of Icosahedron. I have shown that $G$ acts transitively on $G/G_s$ where $G_s$ is the isotropy group, namely ...
1
vote
2answers
66 views

Normal subgroup created by a bunch of elements

if I have a finite group $G$ and a bunch of elements that are the elements of a set $A$. How can I systematically calculate the smallest normal subgroup of $G$ that contains $A$? I am rather ...
1
vote
1answer
26 views

Schur Multiplier of general linear group

Ideally I would like to know the Schur multiplier of $Gl(n, F_3)$, but perhaps this is not reasonable to ask. But for a small fixed $n$, this should be known, but i could not find any result when ...
1
vote
1answer
38 views

Is my proof correct? (about commutators)

I want to prove the following fact: Let $G$ be a finite group and let $x, y\in G$ such that $[x, y] \in Z(G)$. Then $[x, y^s] = [x, y]^s$ for every $s\in \mathbb{Z}$. If we assume that $[x^r, y^s] ...
4
votes
1answer
63 views

Another Presentation of Certain Cyclic Groups

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^n\rangle $$ is cyclic of order $3(n+1)$, for $n=0 \mod 3$ or $n= 1 \mod 3$, $n\ge 0$. This ...
2
votes
2answers
45 views

Determine the center of this finitely presented group.

Consider the group $G(n)= \langle a, b \ \vert\ aba^{-1}=b^{n+1}, bab^{-1}=a^{n+1} \rangle$, $n\ge 1.$ Show that the center $Z$ is cyclic of order $n$ and that $G/Z$ is abelian of order $n^2$. This ...
1
vote
1answer
49 views

Show that the symmetric group $S_p = <\sigma , \tau >$, where $\sigma$ is any transposition and $\tau$ is any p- cycle and p is a prime number.

Let $\sigma = (a_1\ a_2) \ \ and \ \ \tau = (a_1\ b_2\ \ldots\ b_p)$. (We have $a_2 = b_i$ for some i.).We know that $S_p$ is generated by $\{ (a_1\ a_2) \ \ and \ \ (a_1\ a_2\ \ldots\ a_p) \}$. So ...
-2
votes
0answers
37 views

A finite group with some prescribed subgroup structure

Can you display a group $G$ satisfying the following condition? You can choose an element $g\in G$ such that the set $\{H\trianglelefteq G \colon H\ \text{does not contain any power of } g\}$ has ...
0
votes
0answers
24 views

quasiprimitive non-solvable groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
1
vote
4answers
80 views

multiplication in Galois Fields

I don't know much about Galois fields. My question is the following Assume we are working with GF(8). Let say for example I want to multiply 2 by 4 in GF(8). Then it should be equal to $2*4 \text{ ...
6
votes
1answer
51 views

Group elements $x$ and $y$ satisfying $x^2 = y^2x^2y$ and $yx^{-1}y^2 = x^7$ commute.

The Question Suppose that $x$ and $y$ are elements of a group such that $$x^2 = y^2x^2y$$ and $$yx^{-1}y^2 = x^7.$$ Show that $x$ and $y$ commute. Motivation This came up in another question, where ...
3
votes
2answers
99 views
+50

Prove that there is a basis of a lattice $\Lambda$ s.t. a reflection is of a certain form

Consider $\Lambda$ a lattice in $\mathbb{R}^2$. Let $S \in O(\Lambda)$ be a reflection, i.e. $\det S = -1$. Set $S_{1}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$ and $S_{2}= ...
2
votes
1answer
43 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 ...
0
votes
1answer
49 views

what are the other 2 nontrivial elements of the automorphism group of $\Bbb Z/5\Bbb Z$?

It is known that the automorphism group of the units of $\Bbb Z/5\Bbb Z$ is isomorphic to the cyclic group of order $4$, so the automorphism group must also have $4$ elements. The two nontrivial ones ...
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0answers
24 views

Is there a special name for this single element discrete subgroup of Mobius group

Is there a special name for this discrete subgroup of Mobius group with single element: $$A=\begin{bmatrix}0 & i\\-i & 0\end{bmatrix}$$ and $\det A=-1$ and $A^2=I$. Thanks- mike EDIT: ...
4
votes
2answers
203 views

Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle $$ is cyclic of order 24. This presentation was obtained using the Todd-Coxeter ...
2
votes
1answer
32 views

Finite groups with a cyclic maximal subgroup.

In the book A Course in the Theory of Groups by Derek J.S. Robinson, Finite $p$-groups with a cyclic maximal subgroup are classified. Now I wish to know whether finite groups with a cyclic maximal ...
0
votes
0answers
22 views

The number of elements of order $p$ in PSL(n,q)

Let $q=p^n$, where $p$ is prime. What is the number of elements of order $p$ in projective special linear group of degree $3$ that denoted by PSL(3,q)? Is there any method for calculation of the ...
0
votes
1answer
46 views

Inconsistent definition of Sylow p-subgroup

Here is the definition of a Sylow $p$-subgroup from Wikipedia: For a prime number $p$, a Sylow $p$-subgroup (sometimes $p$-Sylow subgroup) of a group $G$ is a maximal $p$-subgroup of $G$, i.e., a ...
2
votes
0answers
67 views

Cokernel of injective endomorphisms of a finitely generated free abelian group

By $\text{GL}^+(n,\mathbb{Z})$ we mean the set of $n×n$ invertible matrices with positive determinant and entries from $\mathbb{Z}$. For given $A \in \text{GL}^+(n,\mathbb{Z})$ let ...
2
votes
0answers
67 views

Groups such that all elements of even order are in $G'$

We know that $G=A_4$ is a group such that all elements of even order in $G$ are in $G':=[G,G]= V_4$, the klein four group. Are there other examples or classes of groups where all elements of even ...
0
votes
0answers
15 views

Prove that both $R_p$ and $R^p$ are abelian groups under ordinary addition of rationals.

Q: Let $p$ be a fixed prime. Let $R_p$ be the set of all those rational numbers whose denominator is relatively prime to $p$. Let $R^p$ be the set of rationals whose denominator is a power of $p\ ...
2
votes
1answer
34 views

Commuting Elements in a Free Product of Cyclic Groups

In the free group with two generators $F_2\cong\mathbb Z *\mathbb Z$ ($*$ denotes the free product), if two elements $a$ and $b$ commute, then there exists an element $w\in F_2$ such that $\langle ...
2
votes
2answers
148 views

Show that $\mathbb{Z}_p\setminus\{\overline{0}\}$ is not a group if $p$ is not prime.

The answer is too short that I think I've gone wrong at some point! Q: If $p$ is prime, then the nonzero elements of $\mathbb{Z}_p$ form a group of order $p-1$ under multiplication. Show that this ...
2
votes
1answer
55 views

Density of nilpotent numbers

A natural number $n$ is nilpotent if every group of order $n$ is nilpotent (equivalently, a direct product of Sylow subgroups). A natural number $n$ has nilpotent factorization if $\ell\not\equiv1$ ...
0
votes
1answer
88 views

(Theorem) If $G$ is a simple group of odd order , then $G \cong \mathbb Z_p$ for some prime $p$.

I am studying Dumit Foote. I have seen this result in this book. Please help me solve this. Thank you.
1
vote
1answer
35 views

A question on the intuition of decomposition of the element of symmetry group

Any element of symmetry group $S_{n}$ can be decomposed as products of transpositions. Any m-cycle can be decomposed as m-1 transposition products. How should I think of this decomposition? Is there ...
2
votes
2answers
58 views

Number of conjugacy classes in finite groups

Let $G$ be a finite group. Let $C_1,C_2,\dots,C_k$ be its conjugacy classes. We denote by $C_{j\ '}=\{g^{-1}|\ g\in C_j\}$ the conjugacy class inverse to $C_j$. Set $$a_{rst} = ...
-1
votes
0answers
36 views

Representations of group algebra and its centre

Are the irreducible representations of the algebra $Z(\mathbb{C}G)$ for a finite group G all irreducible representations of the algebra $\mathbb{C}G$, i.e. are the representations of the group algebra ...
1
vote
1answer
48 views

I need an example of a function with these properties.

I have a problem that says $A$ is finite and $B\subset A$ and that $G$ is the subset set of $S_A$ consisting of all the permutations $f$ of $A$ s.t. $f(x)\in B \ \forall \ x\in B$. these functions ...
0
votes
1answer
56 views

Action of factor group on a group

Suppose $A$ and $G$ are finite groups and $A$ acts on $G$, written $g^a$ for $g\in G$, $a\in A$. If $N\unlhd A$, does $A/N$ then act on $G$ also? By $g^{[a]} = g^a$? Or do I need to assume something ...
1
vote
1answer
47 views

Group theory problem

I am asked to prove "Show that if $${e}<H_1<H_2<...<H_{n-1}<G$$ Is a subnormal series for a group G, and if the order of $H_{i+1}/H_i=s_{i+1}$, then G is of order $s_1 ...