TITLE: Why is it true that if $H$ is a subgroup of a group $G$, then $1_H=1_G$?
QUESTION [2 upvotes]: I am studying about subgroup. My definition of subgroup is that:
Let a set $G$, with a binary operation$
×:G×G→G,(a,b)↦×(a,b)=:a×b$ be a group.　Then $H⊂G$ is a subgroup iff $H$ with a restriction of $×$ to $H×H$, that is,$×|_{H×H}$　is also a group.
And my book states that if $H$ is a subgroup of a group $G$, then $1_H=1_G$. Why is this true?

REPLY [3 votes]: Suppose exists $b\in G,b\ne 1_H$ such that $ab=a$ for some $a\in G.$ By multipling $a^{-1}$ on the left we have that $b=1_G$, but $1_H$ is an element of $G$ which satisies $a1_H=a$ for some $a\in G,$ (in particular, the elements of H). Therefore, $1_H=1_G$.
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