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Questions tagged [groups-enumeration]

Number and enumeration of all finite groups of a given order

The problem of enumerating all finite groups of a given order $n$ means determining complete and non-redundant list of all groups of order $n$, where non-redundancy means that groups from the list are non-isomorphic pairwise.

This tag covers questions about the number of groups of a given order (denoted by $gnu(n)$) and classification of all groups of a given order for various values of $n$. It also includes questions about lower and upper bounds for $gnu(n)$, its asymptotic behaviour, and related questions under certain restrictions on the group.

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More than 99% of groups of order less than 2000 are of order 1024?

In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024. Is this for real? How can one deduce this result? Is there a nice way or do we just check all…
Hui Yu
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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 example. For many $n$, we have $N(n)
Peter
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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 $|a| = p^2; |b|=p;b^{-1}ab = a^{1+p}$ and the other…
Mike
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Known bounds for the number of groups of a given order.

The number of nonisomorphic groups of order $n$ is usually called $\nu(n)$. I found a very good survey about the values. $\nu(n)$ is completely known absolutely up to $n=2047$, and for many other values of $n$ too (for squarefree n, there is a…
Peter
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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 it is an open question, are there any famous…
Izzhov
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How many different groups of order $15$ there are?

I wanted to share with you my resolution of this exercise. How many different groups of order $15$ there are? My resolution: We're looking for groups such that $|G|=15=3\cdot 5$. Then: $G$ has an unique Sylow $3$-subgroup of order $3$ ($P_3$) and…
Relure
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How many Groups there are on a finite set?

Let say cardinality of set S is $n=|S|$. We know that there are $n^{n^2}$ all binary operations on that set. To find out how many groups can be created by this set and by those operations, we need not only to know how many associative operations…
IremadzeArchil19910311
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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 following: $$A(16)=\frac{5}{9}=0.5556,…
Chuks
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Semidirect product uniqueness argument for classifying groups of small order

I'm having trouble understanding the following method for determining the number of semidirect products between two groups in simple cases that arise when trying to classify certain groups of small order. Here is an example, but I'm really…
Otis
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Classify groups of order 27

Let $|G|=27$. Prove that all subgroups of index $3$ are normal. Classify all groups of order $27$. I can do the first one, but the classification is overwhelming. I don't even know where to start. Any help appreciated.
Benjamin Lu
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Reference request: groups of order $p^4$.

I am looking for a textbook or a paper which include the classification of groups of order $p^4$ ($p$ is prime) using generators and relations. In particular I like to understand which group $G$ "exist" for the odd $p$'s and do not exist for $p=2$.
Ofir Schnabel
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Is it "often known" how to compute a list of groups?

From the introduction of the article Construction of Finite Groups written by Hans Ulrich Besche and Bettina Eick: When attempting to determine up to isomorphism the groups of a given order it is often known how to compute a list of groups of…
Jawad
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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 not give a result. The number $n=2058$ is…
Peter
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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 $n_3\equiv 1\text{ mod }3$, hence $n_3 = 1\text{ or…
Zhulin Li
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Groups of order $p^5$

I am reading a paper "A Determination of order $p^5$" by H A Bender ($p$ is an odd prime). He divides the classification in two classes, one which contains an abelian subgroup of order $p^4$ and other do not. He assumes an element of order $p$ which…
Steve
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