Define oentre of the group and prove that it forms a subgroup

The center of a group is defined as the set of all elements that commute with every other element in the group. In other words, if G is a group, the center of G, denoted as Z(G), is defined as:

Z(G) = {a ∈ G | ax = xa for all x ∈ G}

Now, to prove that the center of a group forms a subgroup, we need to show that it satisfies the three properties of a subgroup:

1. Closure: If a and b are in Z(G), then their product is also in Z(G). This is because if ax = xa for all x in G and bx = xb for all x in G, then for the product ab, we have (ab)x = a(bx) = a(xb) = (ax)b = (xa)b = x(ab) for all x in G. Hence, ab is in Z(G).

2. Identity: The identity element of G is in Z(G) because the identity element commutes with all elements of G.

3. Inverses: If a is in Z(G), then its inverse is also in Z(G). This is because if ax = xa for all x in G, then for the inverse a^-1, we have a^-1x = x(a^-1) for all x in G. This is because a^-1x = a^-1(axa^-1) = (a^-1a)xa^-1 = exa^-1 = xea^-1 = xa^-1 for all x in G. Hence, a^-1 is in Z(G).

Therefore, the center of a group forms a subgroup.