INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

September 12, 2013

Center of group and normal subgroup of order 2

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

True

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).

Some Useful Links:

Our College Mathematics Program

Cyclic Group - TIFR Problem - Video

2 comments on “Center of group and normal subgroup of order 2”

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
enter