Question

Let H be a subgroup of a group G. If $e_G$is the identity element of G and e_H is the identity element of H, prove that e_G= e_H

Short answer

It is proved that e_G= e_H

Step-by-step solution

Definition of identity element

An identity element of a group is an element in that group. When the binary operation of group is applied on the identity element with any other element of the group, the other element does not change.

For example, suppose e is an identity element of group G and a is any other element in group G. Then,

$a*e=e$

Proving that  eG = eH

Suppose ais an element of group H.

Since e_H is an identity element of group H

So, $a*e_H=a$……………. (1)

As H is a subgroup of group G, then each element of H is also an element of G. As a corollary,a is also an element of G.

Since e_Gis an identity element of group G

So, $a*e_G=a$……………….(2)

Comparing equations (1) and (2), we get,

$a*e_H=a=a*e_G$

Hence, $e_G=e_H$