[離散]證明subgroup的問題

Let G be a group and let a 屬於 G.
The set C(a)={x 屬於 G | xa=ax} for all
elements of G that commute with a.
Show C(a) is a subgroup of G.

ie.證
1.C(a) 不為空集
2.C(a) is closed
3.C(a) has the identity e
4.C(a) has the inverse property
5.C(a) is associative

我已經知道

1.
let a 屬於 G ,
因為aa = aa => a 屬於 C(a) 不為空集

2.closed:
(先證b = a之inverse 屬於 C(a) )
因為 b = a之inverse 屬於 G and
ba = e = ab
=> b 屬於 C(a)
(再證closed)
任意 x , y 屬於 C(a)
xa = ax , ya = ay
=>xab = axb , yab = ayb => x = axb , y = ayb
=>xy = axbayb
=>xy = axyb
=>xya = axy => (xy)a = a(xy)
=>xy 屬於 C(a)

3.
因為 ea = ae => e 屬於 C(a)

4.
inverse不會證

5.
C(a) inherits the associative property from G
=>G is associative


請問怎麼證有inverse property？