Questions tagged [groups-enumeration]

Number and enumeration of all finite groups of a given order

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7
votes
1answer
128 views

The $221$ groups of order $|G| = 400$

A book of John Conway suggests there are 221 groups of order $|G| = 400$. How do I go about finding these. Commutative groups with $ab = ba$ can be listed very easily: $\mathbb{Z}/400\mathbb{Z}$ $\...
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3answers
99 views

The number of groups of order 32

There are 51 groups of order ($32=2^5$). My question is how this number was computed. Graham Higman and Charles Sims gives an estimate for the number of $p$-goups (i.e. groups of order $p^n$ where $p$-...
2
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1answer
53 views

Classification groups of order 6 and 10

I have this approach to classify groups of order $6$ and $10$. Let $G$ be a group with $|G|=n$ If $n=6$, then by Sylow there is an element $g$ of order $3$. Hence $N:=\langle g\rangle$ has index $2$ ...
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2answers
99 views

Characterization of groups of order 110 (understanding semidirect product)

I try to characterize all groups $G$ of order $110=2 \cdot 5 \cdot 11$. Sylow implies the existence of a subgroup $N \trianglelefteq G$ of order $55$ and a subgroup $H$ of order $2$. It follows, that $...
1
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1answer
131 views

How many (non-isomorphic) groups of order 315 are there?

Show that all such groups are a direct product of a group of order 5 with the semi direct product of a group of order 7 with a group of order 9. Please help me fix my solution! Help appreciated. The ...
0
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2answers
159 views

Conjectures around number of subgroups of symmetric group [closed]

I am asking if you know unsolved or recently solved conjectures around numbers of subgroups in symmetric or alternating groups. In fact, is there a formula depending of $n$ to count subgroups of order ...
1
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1answer
103 views

How to approximate the number of groups?

How can I find a reasonable approximation for the number of groups upto isomorphism of some higher order $\ n\ $ with relatively large exponents in the prime factorization without excessive ...
5
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1answer
595 views

Classify the groups of order $88$ up to isomorphism.

Classify the groups of order $88$ up to isomorphism. Here is what I have so far (I'm aware that there are $12$ groups, but I don't know which ones I'm missing as well as why the $3$ groups are ...
2
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1answer
164 views

Classify all groups of order $375$

According to A000001, the number of groups of order $375$ is $7$. How to figure them out? $5$ groups can be found in this way: Because the number of $125$-order groups is $5$, we can make the direct ...
3
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0answers
292 views

Find all the groups of order 28

I've seen there is a similar question, but it looks for something more specific. I'll write what I thought. I've rapidly seen that every group of order 28 has a unique normal subgroup of order 7. ...
14
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2answers
2k views

Groups of order $p^3$

The following is exercise 8 (section 2.6) in Algebra by Hungerford: Let $p$ be an odd prime. Prove that there are at most two nonabelian groups of order $p^3$. (One has generators $a,b$ satisying $|...
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1answer
232 views

Flaw in my classification of groups of order 2015

In an attempt to classify groups of order $2015 = 5 \cdot 13 \cdot 31$, I deduced that only $\mathbb{Z}/2015 \mathbb{Z}$ was the only such group. I then checked with some sources that informed me that ...
14
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1answer
257 views

Does the list of “number of groups of order $n$” contain every natural number?

In other words: For every natural number $m$, does there always exist an $n$ for which there are exactly $m$ groups of order $n$ up to isomorphism? Or is this an open question in mathematics? If ...
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1answer
517 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
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1answer
610 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
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0answers
185 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 ...
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1answer
2k 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
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1answer
79 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
711 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 ...
3
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0answers
176 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
173 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
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2answers
113 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
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0answers
77 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
90 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
136 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 ...
2
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1answer
188 views

Classifying groups of order 6

I'm trying to proof that if a group $G$ has order $6$, then it is either $\mathbb{Z}_{6}$ or $S_{3}$. I know that there are a lot of solutions to this on the internet, but I want to know why I found ...
1
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0answers
76 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
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1answer
177 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
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1answer
96 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
292 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
98 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
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1answer
237 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
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1answer
199 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
409 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: "$b^2$ ...
4
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1answer
4k 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 ...
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0answers
49 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 $256$ ...
2
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1answer
322 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 ...
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3answers
2k 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
56 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
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1answer
79 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
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1answer
174 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
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3answers
261 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
390 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
61 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
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1answer
199 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
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2answers
185 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
322 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
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0answers
97 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
460 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
340 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 ...