Addition table for Z2 ⊕ Z2 and proving when a² = e implies abelian group

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q1) write out an additiontable for Z2 [tex]\oplus[/tex]Z2


Q2)if a^2=e [tex]\forall[/tex] a in G then g is abelian (solve)
 

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Are you having trouble with (q1)?

For (q2) think of the fact that since a*a=e, for all a in G, a is it's own inverse.
 
please clearlly more
i don't under stand
 
first qustion i don't understand it?
 
What do elements look like for your first group?
 
For the 2nd question a^2 = e implies that a = a^-1 now we want to show that ab = ba for all a, b \in G. So ab = (a^-1 b^-1) from our given but a^-1 b^-1 = (ba)^-1. But we also know that every element is its own inverse and since it's a group we have closure therefore if we call ba = c, then c = c^-1 or in other words ba = (ba)^-1. Now look at what we have ab = a^-1 b^-1 = (ba)^-1 = ba. It's been awhile since algebar, anyone confirm/deny?
 
hashimcom said:
q1) write out an additiontable for Z2 [tex]\oplus[/tex]Z2


Q2)if a^2=e [tex]\forall[/tex] a in G then g is abelian (solve)
First, do you understand that you are expected to make some effort of your own and show your work so we will know what kind of suggestions to make?

For example, what are the members of Z2? And then what are the members of [itex]Z_2\oplus Z_2[/itex]?
 
now, how to proof that if G is group then
o(ab) = o(ba)
o(cac^-1) =o(a)

for all a,b,c in G
note that G not abelian??
 
Your first question hasn't been answered and you have not answered the hint question that people asked you. Halls is correct, YOU have to show some work that you've done on the problem. Do so.
 
Q3)if Gis afinite group of even order, then G contains an element (a) not equal zero s.t.
a^2= e
 
Q4) LET G is abilian of order pq , with g.c.d (p,q)=1 , assume there exist a,b in G and
o(a)= p,o(b) = q show that G is cyclic
 
What part of stop posting questions and start showing your own work was misinterpreted? It is unlikely you will get help if you only keep posting questions.
 
mr NoMoreExams

these question i want to solve it for my homework
and for my searching ...
its very important for me
 
If they're your homework problems, then shouldn't you be the one who solves them?

This forum is not a place where you can get your work done for you. However, if you show some effort, then people will help you out.
 
hashimcom said:
?

I am not sure what's confusing you. This is not a forum to get answers and/or have others do your homework for you. You should ask the question, which we figured out you can do, and then show what YOU would do to solve it. YOU have to show effort, since they are YOUR problems. WE, on the other hand, will offer you hints, suggestions, etc. but YOU should be the one doing the research by reading your book, going to your library, etc. and learning. At the very least you should know definitions which it does not seem you do or are reluctant to post them for some reason.