Answer to Question #195915 in Abstract Algebra for Saba Umer

Question

Let G be a subgroup of GL₂ (Z₄) defined by the set {[m b ,0 1}] such that b ∈ Z₄ and m=±1. Show that G is isomorphic to a known group of order 8?

Answer 1

Given

G be a subgroup of GL₂ (Z₄) 

Define as

$$G={\begin{bmatrix} m & b \\ 0 & 1 \end{bmatrix}}$$

Such that b belong to Z₄ and m=±1.

Now we Show that G is isomorphic to a known group of order 8.

Let known group of order 8 is Z₂ × Z₄

Show to isomoriphic ,then we will be show one one , onto , and morphism.

Define a map

f: G to Z₂ × Z₄

$$f(\begin{bmatrix} m & b \\ 0 & 1 \end{bmatrix})$$

=(m,b)

"One one"

If we take any 2 by 2 matrix such as above then

If both element of 1 st row is same then mapping is same.

It show f is one one.

"Onto"

Each elements belong to Z₂ ×Z₄

There exist a matrix in G .then we say that it is onto.

"Homomophism"

f(aA+bB)=af(A)+bf(B)

Because we know that matrix alsway show the prppery of homomorphism.

So finally we say that G os isomorphic to known group of order 8.