8. Let 𝐺 be a group ith identity 𝑒 suoh that 𝑎2 = 𝑒 for all 𝑎 ∈ 𝐺, then 𝐺 is abelian.

Let 𝐺 = [𝑎 00 0] : 𝑎 𝑖𝑠 𝑎 𝑛𝑜𝑛 − 𝑧𝑒𝑟𝑜 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 } . hho that 𝐺 is a oommutativegroup under multiplioation

Let (A,\bullet) be a monoid sush that for every x in A, x\bullet x= where e is the identity element. Show that (A, ) is an abelian group.

Let 𝑄∗ be the set of non-zero rational numbers and * be the binary operation on Q definedby 𝑎 ∗ 𝑏 = 𝑎𝑏/3. Prove that 𝑄∗ is group under *.

f nth 𝑛th  root of unity be 1,a1,a2,…an−1,1,𝑎1,𝑎2,…𝑎𝑛-1, then ∑n−1r=112+ar∑𝑟=1𝑛-112+𝑎𝑟 is equal toAn⋅2n−12n−1−1

Let f be a function such that 𝑓(−𝑥) = −𝑓(𝑥)  for all x. If ∫02𝑓(𝑥)𝑑𝑥=5, then ∫−22(𝑓(𝑥)+6)𝑑𝑥=Responses661616242434