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Questions tagged [groups-enumeration]

Number and enumeration of all finite groups of a given order

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1answer
232 views

How to classify groups up to an isomorphism?

I have a question asking me to classify all groups of size 6 and below up to isomorphism. I can easily find out the answer, but what I'm interested in is how would one go about classifying all groups ...
4
votes
1answer
313 views

What is the smallest positive integer $n$ such that there are $m$ nonisomorphic groups of order $n$? [closed]

This following question given in Gallian's algebra text: What is the smallest positive integer $n$ such that there are two nonisomorphic groups of order $n$? The answer for this question given in ...
3
votes
0answers
135 views

Understanding classification of finite groups order < 10

I'm interested in understand the classification of finite groups of order less then 10. We have of course that for the Lagrange Theorem all groups of prime order such as $2,3,5,7$ are cyclic and ...
1
vote
1answer
1k views

Classify all groups of order 66, up to isomorphism

how to classify all groups of order 66, up to isomorphism? Firstly, we have 66=$3\times 11\times 2$, suppose the number of the Sylow-11 group of this group is $n_k$, since 11 divides $n_k-1$ we can ...
3
votes
1answer
55 views

Number of groups of order n as a series coefficient

Consider the sequence A000001 in oesis.org: $ g_{n}= $ number of (isomorphism classes of) groups of order n. Is it known for which $ z $ the generating function $ \sum_{n=1}^{\infty}g_{n}z^{n} $ ...
4
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0answers
394 views

Classification of non abelian groups of order $p^3$.

This is not a duplicate of this post, neither of this: they don't give an explicit description of these groups, but only some of their properties. Using GAP to find all the non-abelian groups of ...
2
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0answers
135 views

GAP: Going Beyond The Small Groups Library

So far when using GAP to find examples/counter-examples, my work flow tends to be to load the small group library for a particular order, and then 'filter' through it to see if what I am looking for ...
4
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0answers
125 views

How many groups of order $2^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot 11^2\cdot 13^2$ exist?

The calculation of the number of groups of order $$2^2\cdot 3^2\cdot 5^2\cdot 7^2\cdot 11^2$$ (result $81883$) takes already two hours with GAP. So, the calculation of the number of groups of order ...
3
votes
2answers
80 views

Is a formula for $gnu(2pq^2)$ known, where $q=2p+1\ $?

Let $p$ be an odd prime such that $q:=2p+1$ is also prime. Denote $g(p):=gnu(2pq^2)$ = number of groups of order $2pq^2$ upto isomorphy. The following table shows the first few values ...
2
votes
0answers
70 views

How can we show $gnu(8892)=gnu(9324)$ by hand?

With GAP it can be verified immediately that there are $303$ groups of order $8892=2^2\cdot 3^2\cdot 13\cdot19$ and also $303$ groups of order $9324=2^2\cdot3^2\cdot 7\cdot 37$. I do not expect that ...
3
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0answers
80 views

Smallest number $m$ with $gnu(m)=2017\ $?

$gnu(n)$ denotes the number of groups of order $n$ upto isomorphy. $moa(n)$ denotes the smallest number $m$ with $gnu(m)=n$ $$m=259,083,319,343,897,905=5\cdot 2011\cdot 24133\cdot 1067692187$$ ...
3
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2answers
84 views

relation of classification of finite group and finite simple group.

I know that classification of finite simple group is completed. From the fact, can we say that classification of finite group is completed? I know a few relations of finite group and finite simple ...
1
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0answers
71 views

What is the smallest number $n$ with $gnu(n)=2016$?

$gnu(n)$ denotes the number of groups of order $n$ upto isomorphy. What is the smallest number $n$ with $gnu(n)=2016$ ? The numbers upto $n=2047$ do not satisfy the given property. I do not know ...
1
vote
1answer
114 views

What is $gnu(18,480)\ $?

Probably, the number of groups of order $18,480$ can only be determined with GAP. But I may be wrong, so the question should not be understood to be purely computational. A better lower bound than my ...
1
vote
1answer
87 views

What is the next term in the jumping champions?

I am interested in the jumping champions of the number of groups of order $n$, where $n$ is cubefree. After calling LoadPackage("cubefree"); GAP gives the ...
1
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0answers
282 views

Upto which prime is the calculation of $gnu(2^4\cdot3^4\cdot…\cdot p^4)$ feasible?

The determination of $gnu(n)$, the number of groups of order $n$ upto isomorphism, is very hard in general. But if no large powers are involved, it should be possible for relatively large numbers. ...
0
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1answer
77 views

Number of groups of order $512$ with exponent $2,4,8,16,…$

I want to determine the number of groups of order $512$ with exponent $2,4,8,16,32,64,128,256$ and $512$ The first $500,000$ groups in GAP give the following result : ...
2
votes
1answer
173 views

Sharp bounds for the number of groups of order $75600$?

How can I get sharp bounds for $gnu(75600)$, the number of groups of order $75600$. I tried to determine the number of groups of order $15120$ to get a reasonable lower bound, but I quit GAP after ...
3
votes
1answer
192 views

Number of $p$-groups with order $p^k$ and $|Z|=p$

What is known about the number of groups of order $p^k$ with $|Z|=p$ , which I denote $N(p^k)$ ? For $k=1$ , it is clear, that we have $N(p^k)=1$ : We have one group of order $p$ and since it is ...
0
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1answer
136 views

Classification of groups of order 8

In this document, there is a classification of all groups of order 8: http://www2.lawrence.edu/fast/corrys/Math300/8Groups.pdf I understood it all until the part in the third page that says: "$b^2$ ...
3
votes
1answer
1k views

classify groups of order $36$

Question is to classify all groups of order $36$ I do not even know if it is of my level. Let me try this. Sylow theorem says that there are sylow $2$ subgroups of order $4$ and sylow $3$ subgroups ...
1
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0answers
45 views

Are there $5$ non-cyclic groups of order $2^n$ with an element of order $2^{n-1}\ $?

There are $5$ non-cyclic groups of order $128$ with an element of order $64$, namely $$C64 \times C2\ ,\ C64 : C2\ ,\ D128\ ,\ QD128\ ,\ Q128$$ Similar, there are $5$ non-cyclic groups of order $256$...
1
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1answer
125 views

Classifying groups of order $6$ using semidirect products

Let G be a group of order 6. I am able to do the exercise without semidirect products($G \cong Z_6 $ or $S_3$) but I don't know how to use semidirect products to do this. By Sylow's theorem, there ...
-1
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3answers
712 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?
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0answers
50 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 ...
4
votes
1answer
57 views

Is it known whether all prime powers $p^k$ with $k\ge 8$ are group-abundant?

Denote the number of groups of order $n$ by $gnu(n)$. A natural number $n\ge 1$ is called group-abundant, if $gnu(n)>n$, group-perfect, if $gnu(n)=n$ and group-deficient, if $gnu(n)<n$. I ...
2
votes
1answer
167 views

Is there a typo in the formula or does my GAP-package sglppow fail?

This site deals with group formulas for prime powers $p^k$ for $k\le 7$. The formula for $k=7$ seems to be wrong. I compared the results with GAP and the formula is off by $2453$ for $p=13,17,19,23$ ...
6
votes
3answers
188 views

Number of groups of order $9261$?

I checked the odd numbers upto $10\ 000$ , whether they are group-perfect ($gnu(n)=n$ , where $gnu(n)$ is the number of groups of order $n$), and the only case I could not decide is $$9261=3^3\times ...
5
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1answer
208 views

Non-isomorphic groups with identical structure-description

I constructed the non-abelian groups of order $16$ and listed the structure descriptions. The result was : ...
3
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1answer
51 views

Number of $p$-groups of small order and of exponent $p$

In a very recent paper of MR Vaughan-Lee, it is proved that the number of $p$-groups of order $p^8$ and exponent $p$ is a polynomial (of fourth degree) in $p$. Let us consider $p$-groups of order $&...
2
votes
1answer
141 views

How many groups of order $815,409=3^2\times 7^2\times 43^2$ are there?

I aborted GAP after some hours. I wanted to approve my conjecture that $gnu(n)<n$ for all cubefree numbers $n>1$, where $gnu(n)$ is the number of groups of order $n$, but the case $p^2\times q^2\...
3
votes
2answers
126 views

How many groups of order $2500$ are there?

I aborted the GAP-calculation of $Size(ConstructAllGroups(2500))$ after about $3$ hours. $gnu(2500)$ seems to be a very hard case. Does anyone know $gnu(2500)$ (The number of groups of order $2500$)...
4
votes
2answers
218 views

How can I calculate $gnu(17^3\times 2)=gnu(9826)$ with GAP?

I tried to calculate the number of groups of order $17^3\times 2=9826$ with GAP. Neither the NrSmallGroups-Command nor the ConstructAllGroups-Command work with GAP. The latter one because of the ...
2
votes
0answers
76 views

Is there an efficient method to decide whether $gnu(n)<n$ , $gnu(n)=n$ or $gnu(n)>n$?

Denote : $gnu(n)$ = number of groups of order $n$ It is much easier to decide whether a natural number $n$ is group-deficient ($gnu(n)<n$) , group-perfect ($gnu(n)=n$) or group-abundant ($gnu(n)&...
6
votes
1answer
254 views

Is $gnu(2304)$ known?

I wonder whether the number of groups of order $2304=2^8\times 3^2$ is known. GAP exited because of the memory. $gnu(2304)$ must be greater than $1,000,000$ because of $gnu(768)=1,090,235$ and $768=2^...
8
votes
3answers
263 views

How many groups of order $2058$ are there?

I tried to calculate the number of groups of order $2058=2\times3\times 7^3$ and aborted after more than an hour. I used the (apparently slow) function $ConstructAllGroups$ because $NrSmallGroups$ did ...
4
votes
2answers
182 views

For which cases with $2$ or $3$ prime factors do formulas for $gnu(n)$ exist?

Denote : $gnu(n)=$number of groups of order $n$ For squarefree $n$, there is a closed formula for $gnu(n)$. Prime powers upto $p^7$ are also completely solved and I found a formula for the case $p^2q$...
3
votes
0answers
108 views

What is the smallest squarefree number $n$ with $gnu(n)=79$?

I am searching the smallest squarefree number $n$ with $gnu(n)=79$. $n$ must have more then $3$ prime factors because $p+2\ne 79$ and $p+4\ne 79$ for every prime $p$. The maximum possible number of $...
10
votes
1answer
235 views

Proportion of nonabelian $2$-groups of a certain order whose exponent is $4$

Let $$\displaystyle A(n)=\frac{\text{number of nonabelian 2-groups of order $n$ whose exponent is }4}{\text{total number of nonabelian 2-groups of order $n$}}.$$ Using GAP, I could observe the ...
4
votes
1answer
159 views

Is $n=8{,}574{,}796{,}230$ the smallest squarefree number $n$ with $gnu(n)>10^6$?

The number of groups of order $n$ (gnu(n)) can be calculated by a closed formula, if $n$ is squarefree. I could calculate the values with GAP, additionally, I programmed a version in PARI/GP, working ...
6
votes
0answers
106 views

Is $gnu(n)<n$ always true for cubefree $n>1$?

Let $gnu(n)$ be the number of groups of order $n$. If $n$ is cubefree, (there is no prime $p$ with $p^3|n$), does the inequality $gnu(n)<n$ always hold for $n>1$ ? According to GAP, upto $50,...
4
votes
1answer
471 views

Classify groups of order 2015

How many groups are there of order 2015? Why? I've found that if the group is abelian it is the cyclic group of order 2015. If not, I've tried applying Sylow's theorems and found that there are ...
3
votes
1answer
151 views

Are there cube-free numbers $n$, for which the number of groups of order $n$ is unknown?

For squarefree $n$, there is a formula allowing to compute the number of groups of order $n$. I do not think that such a formula exists for cubefree numbers. If a cubefree number $n$ has the property,...
4
votes
1answer
113 views

GAP Most efficient way to check multiple properties of a group in the small group library

In GAP I would like to search the small groups library looking for groups with specific properties (I suppose this is the most common usage). If I have a list of properties I want to test, what is ...
3
votes
1answer
117 views

Where can I find the known values for the number-of-groups-function upto $10,000\ $?

OEIS shows the number of groups of order $n$ upto $2047$. The Magma-online-calculator uses a database, but already for $1024,2004,2016,...$ it cannot determine the number of groups. Maple seems to ...
25
votes
1answer
542 views

Are there $n$ groups of order $n$ for some $n>1$?

Denote $N(n)$ : the number of groups with order $n$. Can $N(n)=n$ hold for some $n>1$ ? I checked the OEIS-sequence as well as the squarefree numbers $n$ in the range $[2,10^6]$ and found no ...
3
votes
1answer
680 views

Classification of groups of order 12

I have a lot of difficulty understanding this proof we went over in class about classification of groups of order 12. Let $G$ be a finite group of order $n=p^rm$ where $m \nshortmid p$. Denote $...
2
votes
1answer
2k views

Classifying all groups of order 10

I am trying to classify all groups of order 10 upto isomorphism. Suppose $G$=10. Using Sylow's theorem, if $n_p$ is the number of the p-Sylow groups (p prime), then $n_2|5$, so it is 1 or 5. Similarly,...
2
votes
0answers
192 views

Upper bound on groups of order 60

I am aware of the fact that there are 13 non-isomorphic groups of order 60 but the proof of this is really long and something that I cannot present in a few minutes. Hence, I need to give a short ...
8
votes
0answers
409 views

Classify groups of order 171

This is a problem from Stanford Algebra Qualifying Exam, Fall 1998. I know the standard way is to use Sylow theorems and semidirect product. $171 = 9\cdot 19$. By Sylow theorems, $n_3|19$ and $n_3\...