Please give me the justification Let G and G\' be groups, a G. and h : G rightar
ID: 2986326 • Letter: P
Question
Please give me the justification
Let G and G' be groups, a G. and h : G rightarrow G' a group homomorphism. Determine whether each of the following statements is true or false. No need to justify. Hint. How is o(a) defined? Keywords: least positive integer, identity, homomorphism. Extra Credit. (1 point, no partial credit). Is it possible to have groups G and G', a G and a homomorphism h : G rightarrow G'' such that o(a) = 4 and o(h(a)) = 3? Hint. Give a concrete example, or rule out the possibility with a rigorous argument.Explanation / Answer
1. True | a^1=e if and only if a=e
2. True | Since order of a divides 7 as a^7=e and order of a !=1 (by part 1)
3. False | Counter example: in Z8 with addition as operation (2+2+2...(8 times)...) = 0 but (2+2+2+2)=0 so order of 2 is not equal to eight.
4. False | Counter example: G=Z8=G', a=2, h is the trivial homorphism. o(a)=4 but o(h(a))!=4 since h(a)=0
5. False | Counter example: G=Z8=G', h(a)=2a, then o(h(1))=o(2)=4, but o(1)=8