Good day all,
Wikipedia states: There are 2328 groups of order 128 up to ismorphism. How is this calculated? Also, what does "up to isomorphism" mean?
(I know what an isomorphism is...i'm just unsure of the phrasing "up to...") Thanks!
Good day all,
Wikipedia states: There are 2328 groups of order 128 up to ismorphism. How is this calculated? Also, what does "up to isomorphism" mean?
(I know what an isomorphism is...i'm just unsure of the phrasing "up to...") Thanks!
The groups $(\Bbb Z/5\Bbb Z)^\times$ and $\Bbb Z/4\Bbb Z$ are different, but they are isomorphic. If you literally counted all the groups of a given order, that would be like counting sets of a given order - given any cardinal numbers $\kappa$ and $\lambda$ one can exhibit $\kappa$-many sets of size $\lambda$ that are all different, so this is a pointless question to ask because there is no limit to how many sets of some size there can be. Instead, when we count groups we count isomorphic groups as the same.
Anyway, we use the fundamental theorem of finite abelian groups. If $n$ is a whole number with prime factorization $\prod p^e$, and $f(n)$ denotes the number of abelian groups of size $n$, then we can say that $f(n)=\prod f(p^e)$, and moreover that $f(p^e)$ equals the number of integer partitions of the integer $e$. In this case, $128=2^7$ and there are $15$ integer partitions of $7$, so there are $15$ abelian groups of order $2^7$.
Note that $2328$ counts the number of finite groups of order $2^7$, not the number of finite abelian groups of that order. Counting those is very nontrivial.
Up to isomorphism: Groups are sets with an operation. The sets of the groups $\langle a \mid a^2 \rangle$ and $\langle b \mid b^2 \rangle$ are different, so the groups are different. However, the isomorphism $a \leftrightarrow b$ shows that they're isomorphic, hence "the same". That is, one may be made into the other solely through relabelling.
Counting groups can be hard, especially groups of power-of-2 order. See The groups of order 128 by Rodney James, M. F. Newman and E. A. O'Brien, Journal of Algebra, ISSN 00218693, Volume 129,Number 1, Page 136 - 158(February 1990). They used a computer search. The paper seems to be accessible online, so you can read the many, many methods and previous results that went into this computation.