Intersection of subgroups of G is a subgroup of G

Intersection of subgroups of $G$ is a subgroup of $G$ .

Proof. Denote the intersection of subgroups of

G

\begin{align*} K = \cap_{i \in I} H_i \end{align*}

where for all

i \in I

H_i

are subgroups of

G

. First, we will check that

K

is non-empty, i.e., it consists of the identity element. Now, each subgroup of

G

contains an identity element. Intersecting all the subgroups has at least the identity element. Therefore,

K

contains the identity element, and so

K

is non-empty. Secondly, we want to know that if

x,y \in K

, then

xy^{-1} \in K

. Since

x,y \in K

, we have that

x,y \in H_i \ \forall i \in I

. We know that

H_i

is a subgroup of

G

for all

i \in I

. So we get that

xy^{-1} \in H_i

for all

i \in I

. So

xy^{-1} \in K