cmi-entrance 2013 QA2

cmi-entrance · India · pgmath 4 marks Groups True/False with Justification

Let $G$ be a group. The following statements hold.
(a) If $G$ has nontrivial centre $C$, then $G / C$ has trivial centre.
(b) If $G \neq 1$, there exists a nontrivial homomorphism $h : \mathbb { Z } \rightarrow G$.
(c) If $| G | = p ^ { 3 }$, for $p$ a prime, then $G$ is abelian.
(d) If $G$ is nonabelian, then it has a nontrivial automorphism.

Let $G$ be a group. The following statements hold.\\
(a) If $G$ has nontrivial centre $C$, then $G / C$ has trivial centre.\\
(b) If $G \neq 1$, there exists a nontrivial homomorphism $h : \mathbb { Z } \rightarrow G$.\\
(c) If $| G | = p ^ { 3 }$, for $p$ a prime, then $G$ is abelian.\\
(d) If $G$ is nonabelian, then it has a nontrivial automorphism.

Paper Questions

QA1 QA10 QA11 QA12 QA13 QA14 QA15 QA2 QA3 QA4 QA5 QA6 QA7 QA8 QA9 QB1 QB10 QB2 QB3 QB4 QB5 QB6 QB7 QB8 QB9