# Questions tagged [groups-enumeration]

Number and enumeration of all finite groups of a given order

68 questions
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 ...
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 ...
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 ...
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 ...
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 $\sum_{n=1}^{\infty}g_{n}z^{n}$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $gnu(m)=n$ $$m=259,083,319,343,897,905=5\cdot 2011\cdot 24133\cdot 1067692187$$ ...
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 ...
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 ...
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 ...
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 ...
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. ...
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 : ...
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 ...
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 ...
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: "$b^2$ ...
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 ...
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 $256$...
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 ...
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?
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 ...
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 ...
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 $p=13,17,19,23$ ...
3answers
188 views

1answer
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### 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 ...
0answers
106 views

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,...
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 ...
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\...