some abstract algebra help needed

[harbong]

can anyone help me with these abstract algebra problems?

1. Let R+ be the group of positive real numbers under multiplication and let R be the group of all real numbers under addition. Prove or disprove that the map φ: R+ > R defined by φ(x)=4^x is an isomorphism from R+ to R.

2. Use Cayley's Theorum to build an isomorphism map from the additive group Z5 (integers mod 5) to subgroup S3 (symmetric group of permutations).

 

[pka]

1. Let R+ be the group of positive real numbers under multiplication and let R be the group of all real numbers under addition. Prove or disprove that the map φ: R+ > R defined by φ(x)=4^x is an isomorphism from R+ to R.
2. Use Cayley's Theorum to build an isomorphism map from the additive group Z5 (integers mod 5) to subgroup S3 (symmetric group of permutations).

In #1 the mapping is backwards. It cannot be an isomorphism. Isomorphisms map identity to identity. Can that be true of φ? How do you get ‘onto’ the negatives?

For #2, there is a problem with order.
What is the order of Z<SUB>5</SUB>?
What is the order of S<SUB>3</SUB>?

 

[harbong]

Thanks for the help with problem 1!
and for problem 2, I misread my instructor's handwriting (our assignments are written) it is actually from Z<SUB>5</SUB> to S<SUB>5</SUB>...any ideas, it's the last problem on the assignment.

 

[pka]

Z<SUB>5</SUB> is cyclic group generated by 1.
It should easy to find a permutation, t, in S<SUB>5</SUB> of order 5.
Then map Z<SUB>5</SUB> to S<SUB>5</SUB> by φ(n)=t<SUP>n</SUP>.

You can use:
\(\displaystyle \L
\left( {\begin{array}{ccccc}
1 & 2 & 3 & 4 & 5 \\
3 & 5 & 2 & 1 & 4 \\
\end{array}} \right)\)