[math-fun] Simple finite group problem
This is a simple finite group problem, not to be confused with a finite simple group problem. It's a simple problem about finite groups, not a difficult problem about finite simple groups. List all multiplicative groups of integers mod ten. For instance {1,9} is one of them, and {1} is another. I noticed that the number of multiplicative groups of integers mod N is not in OEIS. I plan to add it.
If I understand the problem, it's to list all subgroups of the group of invertible elements of the ring Z/10. The invertible elements can be chosen as {-3, -1, 1, 3}. Since there is an element of order 4, this is isomorphic to the abelian group Z/4, so it has: * 1 subgroup of size 4, * 1 of size 2, * 1 of size 1: {-3, -1, 1, 3}, {-1, 1}, {1}. —Dan
On Apr 19, 2016, at 8:06 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
List all multiplicative groups of integers mod ten. For instance {1,9} is one of them, and {1} is another.
Are you counting groups as distinct that are identical to each other, but happen to use different numbers for the elements? For example, {1, 3, 7, 9} is really the same group as {6, 2, 8, 4} (the elements correspond in the order listed). Does it makes sense to count it twice? If so, then we're really identifying subsets of integers that generate these groups, rather than the groups themselves. Tom Keith F. Lynch writes:
This is a simple finite group problem, not to be confused with a finite simple group problem. It's a simple problem about finite groups, not a difficult problem about finite simple groups.
List all multiplicative groups of integers mod ten. For instance {1,9} is one of them, and {1} is another.
I noticed that the number of multiplicative groups of integers mod N is not in OEIS. I plan to add it.
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Dan Asimov -
Keith F. Lynch -
Tom Karzes