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S
UMMARY OF
9/17 L
ECTURE
We introduced several new pieces of notation over the course of the lecture:
H < G
means
H
is a proper subgroup of
G
;
d

n
, read “
d
divides
n
”, means
n
is a multiple of
d
; and

G

, read “the order of
G
”, is the number of elements of
G
(possibly inﬁnite).
We started class with a recap of the proof that every subgroup of
(
Z
,
+)
is of the form
d
Z
for
some natural number
d
.
Next, we turned to a curious application of the theorem. Pick any two integers
a
and
b
, and
consider the set
H
=
{
ma
+
nb
}
m,n
∈
Z
.
This is clearly a subset of
Z
, but we noticed that it’s actually a subgroup. By our theorem, there
exists a natural number
d
such that
H
=
d
Z
. What’s the relationship between
a,b
and
d
? After
playing around with some examples, we ﬁgure out the following:
Claim:
d
= gcd(
a,b
)
Proof:
Step 1:
d

a
and
d

b
.
Observe that
a
∈
H
. Since
H
=
d
Z
, we must have
a
∈
d
Z
, i.e.
a
is a multiple of
d
.
Similarly, we conclude that
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 Fall '10
 GideonMaschler

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