S
UMMARY OF
9/15 L
ECTURE
We started with a definition (motivated by examples).
Let
G
be a group. We say
H
is a
subgroup
of
G
(denoted
H
≤
G
) if:
1.
H
⊆
G
; and
2.
H
is a group
under the same binary operation as
G
.
Thus, although
{
1
,
1
}
is a subset of
Z
, and is a group under multiplication,
(
{
1
,
1
}
,
×
)
is
not
a subgroup of
(
Z
,
+)
. On the other hand, it
is
a subgroup of
(
Q
×
,
×
)
. Do you remember
some of the other examples of subgroups we discussed? Yes or now, it’s a good exercise to look
through our examples of groups from the first two lectures and try to come up with subgroups
of them.
Given a group
G
and a sub
set
S
⊆
G
, we defined the group ‘generated’ by
S
to be the smallest
subgroup of
G
containing
S
. (‘Smallest’ means: every group containing
S
must also contain
the group generated by
S
.) In other words, given
S
, what other elements of
G
would you
have
to add to
S
to make it into a group (under the same binary operation as
G
)? This can
be a somewhat subtle question in general, so we looked at a special case: what is the group
generated by a single element
x
∈
G
? Let’s denote the binary operation of
G
by
@
. We have
to find the smallest subgroup of
G
containing
x
. Well, first of all, to be a group it has to contain
the identity
e
of
G
. Also, to be a group it must have inverses, so
x

1
has to be in there. What
else? Well, any group is ‘closed’ (i.e. satisfies the closure axiom: the combination of any two
elements of the group lands somewhere in the group). So
x
@
x
has to be in there. Similarly,
x

1
@
x

1
has to be in there. What else?
x
@
x
@
x
and
x

1
@
x

1
@
x

1
must be there, etc. In
general, all elements of the form
x
@
x
@
· · ·
@
x

{z
}
n
and
x

1
@
x

1
@
· · ·
@
x

1

{z
}
n
have to be in there, for every positive integer
n
.
Because this notation is getting unwieldy, we came up with a new notation: let
x
n
=
x
@
x
@
· · ·
@
x

{z
}
n
and
x

n
=
x

1
@
x

1
@
· · ·
@
x

1

{z
}
n
for every positive integer
n
. It will also be convenient to write
x
0
=
e
, the identity.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 GideonMaschler

Click to edit the document details