Question

(a) If G is a group and $\text{ab\hspace{0.17em}\hspace{0.17em}∈\hspace{0.17em}\hspace{0.17em}Z}\left(\text{G}\right)$ ,is it true that a and b are in Z(G)? [Hint: D₄ . ]

Short answer

Answer:

If$\text{ab\hspace{0.17em}\hspace{0.17em}∈\hspace{0.17em}\hspace{0.17em}Z}\left(\text{G}\right)$, then a and b does not belong to Z(G) .

Step-by-step solution

Step by Step Solution Step 1: Definition of Dihedral group  D4

A group which is closed under composition 'o' of functions is known to be associative. This group consists of identity element and every element of the group has an inverse. This group is called a dihedral group of degree 4 or the group of symmetries of the square.

The various symmetry mappings of the square are identity mapping e, the rotations r_1,,r_2,r₃ of ${90}^o,{180}^o,{270}^o$ around the center of the square anti-clockwise respectively.

Check whether  a and b are in Z(G)

No, this is not true. If $ab\in Z\left(G\right)then,a,b\in Z\left(G\right)$

In D₄we have:

$Z\left(G\right)=\left\{\left.r_0,r_2\right\}\right.and{r}_1^2=r_2$

Therefore, taking a=b=r₁ gives a²= r₂, and b²=r₂ .

This shows that $ab\in Z\left(G\right)$, and it doesn’t mean that a and b are in Z(G).