Intersection of subgroups is subgroup

This article gives the statement, and possibly proof, of a basic fact in group theory.
View a complete list of basic facts in group theory
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|
VIEW: Survey articles about this

Statement

Verbal statement

The intersection of any arbitrary collection of subgroups of a group is again a subgroup.

Symbolic statement

Let $H_i$ be an arbitrary collection of subgroups of a group $G$ indexed by $i\in I$. Then, ${\bigcap }_{i\in I}{H}_i$ is again a subgroup of $G$.

Proof

Given: Let $H_i$ be an arbitrary collection of subgroups of a group $G$ indexed by $i\in I$. Let us denote $H={\bigcap }_{i\in I}{H}_i$.

To prove: We need to show that $H$ is a subgroup. In other words, we need to show the following:

1. $e\in H$
2. If $g\in H$ then $g^{-1}\in H$
3. If $g,h\in H$ then $gh\in H$

Proof: Let's prove these one by one:

1. Since $e\in H_i$ for every $i$, $e\in H$
2. Take $g\in H$. Then $g\in H_i$ for every $i\in I$. Since each $H_i$ is a subgroup, $g^{-1}\in H_i$ for each $i\in I$. Thus, $g^{-1}\in H$.
3. Take $g,h\in H$. Then $g,h\in H_i$ for every $i$, so $gh\in H_i$ for every $i\in I$. Thus $gh\in H$.

References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 62, Section 2.4 (Subgroups generated by subsets of a group), Proposition 8, (considers the case of a nonempty collection)^{More info}. Also, Page 48, Exercise 10(a) and 10(b) (10(a) asks for the special case where there are only two subgroups being intersected)
• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, Page 3, Proposition 1, ^{More info}
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, Page 8, Section 1.3 (Intersections and joins of subgroups), Proposition 1.3.2, ^{More info}
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, Page 48, 3.3.4, ^{More info}
• Algebra by Serge Lang, ISBN 038795385X, Page 9, ^{More info}
• A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, Page 75, Exercise 54, (only the finite case)^{More info}
• Topics in Algebra by I. N. Herstein, Page 46, Problem 1, (only the finite case, hinting at the infinite case)^{More info}