Question
question 8 ,show step by step ,thank you !
G, TaLbLTa where 74 Groups fined as in Example 8 5. In Problem 4, show that V E A(G) is such thatT 4. In Problem 3 prove that for all a n Problem 3 prove that for all a, b EG, T, L on e E G a E G. then VL, for some b E G. (Hint: Acting n what b should be.) 6. Prove that if : G G, is a homomorphism, then (G), the ima is a subgroup of G if and only if Ker = (e) G', the group of all positive real numbers under multipli IT.,E G a rational], then HG mal subgroup of G, the dihedral group of order 8 G, where is a homomorphism, is a monomor 7. Show that 8. Find an isomorphism of G, the group of all real numbers u 9, verify that if G is the group in Example 6 of Section 1, and 10. Verify that in Example 9 of Section 1, the set H = { i, g, g, g3l is under +,onto 11. Verify that in Example 10 of Section 1, the subgroup s normal in G 12. Prove that if Z(G) is the center of G, then Z(G) G 13. If G is a finite abelian group of order n and : G G is defined by | 25- Thema of G int (b) The im (a)-am for all a E G, find the necessary and sufficient condition that 14. If Gis abelian and : G G, is a homomorphism of G onto G, prove 15. If G is any group, N 16. If N 17. If M 4G, N G, prove that M n N G be an isomorphism of G onto itself. that G' is abelian. G, prove that the image, (N), of N is a normal subgroup of G, a subgroup of G and that MN G if G is G, and : G G, a homomorphism of G onto Middle-Level G and M
Explanation / Answer
8. let f : (G,+) ---> (G', . ) be defined as f(x) = e^x (where G and G' are as defined above).
we need to show f is an isomorphism
HOMOMORPHISM :
let, x,y belongs to G , then f(x+y) = e^ (x+y) = e^x . e^y = f(x) . f(y) . hence f is a homomorphism
ONE ONE :
let f(x) = f(y) in G'. To show : x=y
we have f(x)=f(y) implies, e^x = e^y , i.e. x=y (since x and y are real numbers, exponential is not multivalued)
hence f is one one
ONTO :
let y belongs to G'. then y > 0. hence log(y) is defined. [log is upon base e]
and f(log y) = e^ (logy) = y.
hence for all y in G' there exists x=log(y) in G such that f(x)=y.
hence f is onto.
hence f is an isomorphism
