Answer to Question #107069 in Abstract Algebra for Garima Ahlawat

Question

(1) any two groups of order m are isomorphic where m€ N
(2)if A and B are two sets such that A U B = phi , then A intersection B = phi
True/false. Justify.

Answer 1

1. False.

Consider the group $D_4 =<r,s \mid r^4=s^2=1,rs=sr^{-1} >$.

and $\Z_4=\{0,1,2,3,4,5,6,7 \}$

Where $D_4$ is dihedral group of order 8 and $\Z_8$ is a additive group of integer modulo 8.

Both $D_4 \ and \ \Z_8$ have order 8 but $D_4$ is not isomorphic to $\Z_8$

because $D_4$ is non abelian and $\Z_8$ is abelian.

2.True.

Since $A \cup B=\{ x:x\in A \ or \ x\in B \}=\phi$

Therefore,$A=B=\phi$ .

Hence,$A\cap B=\phi$ .