Mathematics Assignment: Isomorphism, Group Theory, and Proofs

This mathematics assignment delves into the concepts of isomorphism and group theory, offering solutions to problems related to these areas. The first problem demonstrates that the group Z is isomorphic to Z x Z by showing that a function f is one-to-one, onto, and preserves the operation. The second problem proves that a function f: H -> G is an isomorphism by showing that it is onto and one-to-one, and that it preserves multiplication. The assignment provides detailed explanations and proofs to facilitate a better understanding of abstract algebra concepts. This document is available on Desklib, a platform that provides students with access to various study resources including past papers and solved assignments.

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Answer to question 10:
Given that “+” is defined on Z x Z
Therefore, the group Z is said to be isomorphic to Z x Z if
[i] f is one to one and onto
[ii] f((m,n)+(a,b))=f(m,n)f(a,b)
Now,
Let f(m,n) =mn m−1 n−1=1
If x is a rational of the form mn m−1 n−1=1, then f(m,n) = =mn m−1 n−1=1, so f is onto
Let f(m,n) = f(a,b)
Then, mn m−1 n−1=aba−1 b−1
So, mn/ab = a−1 b−1
m−1 n−1
Or, mn/ab =mn/ab =1
Using fundamental theorem of arithmetic, it can be said that (m,n) = (a,b)
Hence, f is one to one.
Finally,
f((m,n) + (a,b))
= f(m+a, n+b)
=(m+a)(n+b)(m+a)^-1(n+b)^-1
=1
=(mn m−1 n−1)¿ aba−1 b−1 ¿
=f(m,n)f(a,b)
Thus, f is operation preserving, so it's an isomorphism.
Hence, Z is isomorphic to Z x Z

Answer to question 11:
Let “*” is the operator for both G and H
Therefore, f: H -> G is said to be isomorphic if:
[a] f is onto and one to one
[b] f((m,n)*(a,b))=f(m,n)f(a,b)
The function f preserves multiplication in H since for all a,b in H we have
f ((m,n)(a,b)) =(mn)^2(ab)^2 = m^2n^2a^2 b^2 = f (m,n) f (a,b)
The function is one-to-one and onto since for each (x,y) in H the equation f (x,y) = x^2y^2
has the unique solution z =sqrt(x^2y^2).
Hence, f is isomorphism.