Isomorphisms and Automorphisms
Study Guide
Outline
1. Isomorphisms
Let
G
and
H
be groups. An
isomorphism
from
G
to
H
is a function
ϕ
:
G
→
H
satisfying
the following conditions:
•
ϕ
is a bijection, and
•
ϕ
(
xy
) =
ϕ
(
x
)
ϕ
(
y
) for all
x,y
∈
G
.
If there exists an isomorphism from
G
to
H
, then the groups
G
and
H
are said to be
iso
morphic
.
Problems:
3
2. Isomorphism Type
If
G
and
H
are isomorphic groups, then:
•
G
and
H
must either both be abelian or both be nonabelian, and
•
G
and
H
must have the same number of elements of each order.
For small groups, this makes it possible to determine the isomorphism type of a given group.
A group of order
n
that has an element of order
n
is cyclic. The following table shows the
isomorphism types of noncyclic groups that we are aware of:
Abelian Groups (Not Cyclic)
Order
Groups
4
V
8
Z
4
×
Z
2
, V
×
Z
2
9
Z
3
×
Z
3
12
Z
6
×
Z
2
16
Z
8
×
Z
2
, Z
4
×
Z
4
,
Z
4
×
V, V
×
V
NonAbelian Groups
Order
Groups
6
S
3
8
D
4
, Q
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 Spring '09
 Algebra, Isomorphism, Cyclic group, Group isomorphism, automorphisms, automorphism group

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