this has to do with direct products
list a group of order 81 with the property hat every element except the identity has order 3
don' even know where to start
2007-11-07 13:05:34 · 2 answers · asked by mlbabe03 2 in Science & Mathematics ➔ Mathematics
The prior answer is correct, although the stated reason makes no sense. Take an element in Z3xZ3xZ3xZ3 and cube it. That cube has to equal the identity. Hence every element has order dividing 3. Hence every non-identity element has order 3.
It's that simple!
Harder is the question as to whether, up to isomorphism, this is the ONLY answer. I'm pretty sure it is, but I'm rusty on my group theory (last took a course on the subject in 1975), and a proof isn't immediately jumping to mind.
2007-11-07 19:17:33 · answer #1 · answered by Curt Monash 7 · 0⤊ 0⤋
i will try but i dont know if it works...
Z3 x Z3 x Z3 x Z3 ... since all the numbers have 3 as their LCM.
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2007-11-07 21:15:52 · answer #2 · answered by Alam Ko Iyan 7 · 0⤊ 0⤋