Theorem 15.1.
If $\cal{C}$ denotes any collection
of subgroups of a group $G,$ then the
intersection of all of the groups in
$\cal{C}$ is also a subgroup of $G.$
Example 15.1.
In $\Z$ (operation addition),
$\gen{3}$
consists of all
the multiples of $3$ and
$\gen{4}$
consists of all multiples of $4.$
Because a number is a multiple
of both $3$ and $4$ iff it is a
multiple of $12,$ we have
\[
\gen{3}\cap\gen{4}=\gen{12}.
\]
Also,
\[
\gen{6}\cap\gen{8}=\gen{24}.
\]
Note.
In the previous example,
note that
$\text{LCM}(3,4)=12$
and
$\text{LCM}(6,8)=24.$
Look ahead to Problem 15.24.
Definition.
If $S$ is any subset of a group $G,$
then $\gen{S}$
is the intersection of all of the
subgroups of $G$ that contain $S.$
Theorem 15.2.
$\gen{S}$
is the unique smallest subgroup of $G$
that contains $S,$ in the sense that
-
$\gen{S}$ contains $S.$
-
$\gen{S}$ is a subgroup, and
-
if $H$ is any subgroup of $G$ that
contains $S,$ then $H$ contains
$\gen{S}.$
Definition.
We say that $S$ generates
$\gen{S}$
and that
$\gen{S}$
is generated by $S.$
If $S=\{a_1,\ldots,a_n\},$
then we write
\[
\gen{S}=\gen{a_1,\ldots,a_n}
\]
rather than
\[
\gen{S}=\gen{\{ a_1,\ldots,a_n \}}.
\]
If $S=\{a\}$ then
\(
\gen{S}=\gen{a}
\)
is just the cyclic subgroup
generated by $a.$
Example 15.2
The subgroup
$\gen{9,12}$
of the group of integers must
contain $12+(-9)=3.$
Therefore,
$\gen{9,12}$
must contain all multiples of $3.$
That is,
\(
\gen{9,12}\supseteq\gen{3}.
\)
But also,
\(
\gen{9,12}\subseteq\gen{3},
\)
since both $9$ and $12$ are
multiples of $3.$
Therefore,
\(
\gen{9,12}=\gen{3}.
\)
The next theorem generalizes
this example.
Theorem 15.3.
If $T_1$ and $T_2$ are subsets of a
group $G,$ then
\(
\gen{T_1}=\gen{T_2}
\)
iff
\(
T_1\subseteq\gen{T_2}
\)
and
\(
\gen{T_1}\supseteq T_2.
\)
Steps.
Use repeated applications of
-
Adjoin to $S$ all elements
$ab$ for $a,b\in S.$
-
Adjoin to $S$ all elements
$a^{-1}$ for $a\in S.$
-
Repeat steps 1 and 2
on the new set obtained
from steps 1 and 2.
-
Stop when no more new
elements are found.
Theorem.
If $S$ is a nonempty subset of $G$ then
\(
\gen{S}=\{ a_1\cdots a_k\mid
k
\)
a positive integer,
\(
a_i\in S
\)
or
\(
a_i^{-1}\in S
\)
for all
\(
i=1,\ldots,k\}.
\)
Example 15.3.
Using Theorem 15.3, verify that
in $S_4,$
\[
\gen{(1\ 2\ 4),(2\ 3\ 4)}
=\gen{(1\ 2\ 3),(1\ 2)(3\ 4)}.
\]
Definition.
If $A$ and $B$ are groups,
then $A\times B$ is the
Cartesian product of $A$
and $B.$
\[
A\times B=\{(a,b)\mid a\in A\text{ and } b\in B\}.
\]
Theorem 15.4.
If $A$ and $B$ are groups, then
$A\times B$ is a group with respect
to the operation defined by
\[
(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2)
\]
for all $a_1,a_2\in A$ and $b_1,b_2\in B,$
where $a_1a_2$ is the operation in $A$
and $b_1b_2$ is the operation in $B.$
The group $A\times B$ (with this operation)
is called
the direct product of $A$ and $B.$
Example.
In $\Z\times\Z$ we have
\[
(a,b)(c,d)=(a+c,b+d).
\]
Theorem.
If $A$ and $B$ are finite, then
so is $A\times B$, and
\[
\abs{A\times B}=\abs{A}\cdot\abs{B}.
\]
Example.
\(
\abs{\Z_m\times\Z_n}
=\abs{\Z_m}\cdot\abs{\Z_n}
=mn.
\)
Example 15.4.
Find the elements of $\Z_3\times S_2$
and compute $ab$ for two elements
$a$ and $b$ in the group.
Theorem.
If $A$ and $B$ are groups, then
\[
A\times\{e\}=\{(a,e)\mid a\in A\}
\]
and
\[
\{e\}\times B=\{(e,b)\mid b\in B\}
\]
are subgroups of $A\times B.$
Alex Notes