U16  is the set of elements in Z16 which has inverse in Z16. Prove that U16 is a group under multiplication in Z16.  PLZ with details

Prove that Z is not a group under usual subtraction -. Explain why (Z,-) is not an abelian group

Construct the group Z4 building an isomorphism. Z8/ and determine a group that it is isomorphic to without Show your work and explanations.

Q1\Find the inverse for each element in the following mathematical systems (Z, *) where * defined as a*b= a+b +11 Va, b €Z. The Kline four group. A) B) C) (D) (E) (Ze, +). (R, *) where * defined as a*b= a+b+ab Va, b ER. (P(X), A) where X-(0, 1, 2). dlic groun has a unique ident

8. Show that (Z,,×s) is a monoid. Is (Z.,×6) an abelian group? Justify your answer 9. List two famous mathematicians and discuss their contributions in mathematics.

Help plz

4. Consider the complex number set { 1,-1,1,-i, (¹+i) (1-i) (-1+i) (-1-i) √√2 √2 √√2 √√2 ordinary multiplication is a group. a) Can you identify a subgroup of it? b) Can you identify a generator for it? } together with

following initial value problems. explicit solution. си Soluc the If possible writo b) 2 dy dx = y − xy, y(−1) = −1