Discussion
Consider the following statements :
I : If G is a group such that every cyclic group of G is a normal subgroup of G, then every subgroup of G is a normal subgroup of G.
II : If H is a subgroup of a group G such that the index of H in G is 2, then H is a normal subgroup of G.
Which of the following statement(s) is/are true ?
Answer: C
Both I and II are true.
I: any subgroup H is the union of the cyclic subgroups \(\langle h \rangle\), \(h \in H\). If each \(\langle h \rangle\) is normal, then \(g h g^{-1} \in \langle h \rangle \subseteq H\) for every g, so H is normal. (Groups like this are called Dedekind groups; the quaternion group is a non-abelian example.)
II: if H has index 2, its two left cosets are H and \(G \setminus H\), and so are its two right cosets. So \(gH = Hg\) for every g, which is normality.
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