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May 17th, 2017, 02:27 PM  #1 
Newbie Joined: May 2017 From: Neverland Posts: 7 Thanks: 0  [Set Theory]  filling tablet
(G, *) is a group of 4 elements, a,b,c,e are different elements of G and e is a neutral element in (G, *). given this equation: a*b = c*c how can i start filling the tablet of (G, *) ? i've understood that the column and row of the neutral element are just the same as the one that goes with it, a*e=a same goes to b,c and e. and then i have this " every element in a tablet of a group can show only once in each row or column. and can i assume that if a*b = c*c then b*a = c*c as well? i'll appreciate any help or clues 
May 17th, 2017, 03:22 PM  #2  
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  Quote:
No, the group isn't Abelian unless the problem says it is. Now in some cases we can assume commutativity. For example if $xx^{1} = e$ then we can conclude that $x^{1}x = e$ as well. But in general you can't assume commutativity.  
May 17th, 2017, 04:27 PM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,960 Thanks: 802 
There are only two groups, up to isomorphism, of order 4, the rotation group and Klein four group and they are both Abelian.

May 18th, 2017, 05:43 AM  #4 
Newbie Joined: May 2017 From: Neverland Posts: 7 Thanks: 0  maybe.jpg [attached a picture] so i've put all the Given one's first, then i start like this: c*a can't be equal to a or c, because that'll make a or c nuetral elements which is wrong (Given that e is nuetral), niether can be a*b cuz it's already exist in the row so c*a = b. and then c*b = a, because it's the only element left (no element duplicating) and a*c = b because, a already exists in the row so it can't be, niether a*b or c (no duplicating) so we're left with b. then can i assume that c*c = a*b = e (nuetral element) ? because Given that G only Has 4 elements and e is the nuetral one within them, and it's the only one left in this row. is that's the way ? 
May 21st, 2017, 07:16 AM  #5 
Newbie Joined: May 2017 From: Neverland Posts: 7 Thanks: 0 
may anyone look up my answer please

May 21st, 2017, 09:49 AM  #6 
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  
May 26th, 2017, 01:48 PM  #7 
Newbie Joined: May 2017 From: Neverland Posts: 7 Thanks: 0 
thanks guys i've got it in the end !


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filling, set, tablet, theory 
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