Center of group and normal subgroup of order 2

Any normal subgroup of order 2 is contained in the center of the group.


Discussion: Center of a group Z(G) is the sub group of elements that commute with all members of the group. A subgroup of order two has two elements: identity element and another element, say x, which is self inverse. Since Z(G) is a subgroup it contains the identity element. We show that the other element x also is in Z(G).

Suppose H is the normal subgroup of order 2 and H={1, x}. If g is an arbitrary element of G, then gH = Hg as H is normal. That is {g, gx} = {g, xg}. Since the two sets are equal and x is not identity, hence gx = xg. This implies that x is in the Center of the Group (as it commutes with an arbitrary element of the group).

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