Answer with Step-by-step explanation:
We are given that a group H=[tex](R+,\times)[/tex] b consisting of the set of all strictly positive real numbers with binary operation given by multiplication.
We have to show that H is isomorphic to (R,+) where (R,+) is a group consisting of real numbers with binary operation given by addition.
Isomorphic group :If there is one-one correspondence between the elements of two group with respect to given binary operations and there exist an isomorphism between two groups then the groups are called isomorphic groups.
Suppose a function
f:[tex](R+,\times )\rightarrow (R,+)[/tex]
f(x\cdot y)=f(x)+f(y)
If two groups have same order and have same order elements then we also says that the groups are isomorphic.
Order of H is infinite and order of (R,+) is also infinite.
H have one element of order 1 and one subgroup of order of order 1 and (R,+) have one element of order 1 and one subgroup of order 1.
(R,+) is infinitely generated and and H is also infinitely generated group.
Both have same properties .Hence , H and (R,+) are isomorphic group.
