[Math] Group tables for a group of four elements.

abelian-groupsabstract-algebrafinite-groupsgroup-theory

I should consider group tables obtained by renaming elements as essentially the same and then show that there are only two essentially different groups of order 4.

There seems to be so many different possible group tables for all the different binary operations – which is why I'm confused. I was thinking about using Cayley's table to show their commutativity but I'm really not too sure. Any help please!

Edit: After all of your help, I completely understand how to show that there are only two different groups of order four. Thank you. The only thing which I am still unclear of is how to note down the 'other tables' before stating that they are essentially the same as one of the other tables – meaning that there are just two different ones. It's pointless work but I think it's what the question requires.

Answer: After a bit of playing around – I've realised that there are four different tables, however, 3 tables are the same as each other, just with different values (the Klein 4 Group with 3 different generators). Hence there are two tables.

Best Answer

Hints: we are talking about groups of order 4, which narrows the possible binary relations and elements available:

  • each must contain the identity,
  • each must be associative,
  • each must be closed under inverses (if $a \in G,\;a^{-1} \in G$)
  • (and of course, closed under the group operation).

There are essentially (up to isomorphism) only two groups of order 4:

  • one of course will be the additive cyclic group $\mathbb{Z}_4$, and
  • the other will be the Klein 4-group, which is indeed abelian.

If you follow the suggestions for solving the problem, you'll find, indeed, that any possible GROUP of order 4 can be shown isomorphic to one of the two groups mentioned simply by a renaming elements.

Yes: use of the Cayley table will be of great importance:

  • for a group: no element can appear twice in any column,
  • no element can appear twice in any row.

You'll find that the only ways of completing a table satisfying these criteria are limited, and then you show that a simple renaming of elements will reveal the group is isomorphic to $\mathbb{Z}_4$ or else the Klein-4 group.

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