Center (group Theory) - As A Subgroup

As A Subgroup

The center of G is always a subgroup of G. In particular:

1. Z(G) contains e, the identity element of G, because eg = g = ge for all g ∈ G by definition of e, so by definition of Z(G), e ∈ Z(G);
2. If x and y are in Z(G), then (xy)g = x(yg) = x(gy) = (xg)y = (gx)y = g(xy) for each g ∈ G, and so xy is in Z(G) as well (i.e., Z(G) exhibits closure);
3. If x is in Z(G), then gx = xg, and multiplying twice, once on the left and once on the right, by x−1, gives x−1g = gx−1 — so x−1 ∈ Z(G).

Furthermore the center of G is always a normal subgroup of G, as it is closed under conjugation.