Union of Subgroups

• Mar 17th 2010, 01:00 PM
jameselmore91
Union of Subgroups
True or False:

If H and K are subgroups of G, then $H \cap K$ is a subgroup of G.
If $H_{t}$ is a subgroup of G for all $t \in I$, then $\cap_{t \in I} H_{t}$ is a subgroup of G.

Explanations would be greatly appreciated.
• Mar 17th 2010, 02:10 PM
Dinkydoe
You only need to verify that these sets are groups,

that is
(1) $e\in K\cap H$
(2) $a\in K\cap H \Rightarrow a^{-1}\in K \cap H$
((3) associativity is automatically satisfied)

These are trivial exercises. Keep the definition of intersections in mind.
The same goes for $\bigcap_{t\in I} H_t$

(Edit: What you wrote was not the right symbol for Union. The union symbol is $\bigcup$. However the excercise is the same)
• Mar 17th 2010, 02:57 PM
Drexel28
Quote:

Originally Posted by Dinkydoe

(Edit: What you wrote was not the right symbol for Union. The union symbol is $\bigcup$. However the excercise is the same)

I disagree. The union of two subgroups need not be a subgroup. For example

$2\mathbb{Z}\cup3\mathbb{Z}\not\leqslant\mathbb{Z}$
• Mar 17th 2010, 03:34 PM
Dinkydoe
Yes, true. However on second thought, the poster mustve meant -intersection of sub-groups-
• Mar 17th 2010, 03:43 PM
jameselmore91
My mistake, I did mean to say intersection of Subgroups