I think they meant it the other way around. They wanted to talk about abelian groups, and here specifically a cyclic (abelian) group. It would be silly to talk

I think they meant it the other way around. They wanted to talk about abelian groups, and here specifically a cyclic (abelian) group. It would be silly to talk about a cyclic group that is specifically abelian, all cyclic groups are abelian. But not all abelian groups are cyclic. They could have kept it implicit, but not everyone might know that fact, so it doesn't really hurt