Theorem 14.1.
Let G be a group and a,b,c∈G.
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If ab=ac then b=c. (left cancellation law)
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If ba=ca then b=c. (right cancellation law)
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ax=b has a unique solution in G, x=a−1b.
xa=b has unique solutions in G, x=ba−1.
-
(a−1)−1=a.
-
(ab)−1=b−1a−1.
Note.
In the Cayley table of G,
ax will be in the row labeled by a.
Note.
If a,x∈G, a finite group, Theorem 14.1
implies ax appears in the row labeled by
a in
G's
Cayley table.
Note.
If b∈G, a finite group, and ax=b,
Theorem 14.1(c) implies b appears
exactly once in the row labeled by a in
G's
Cayley table.
Theorem.
Ignoring row labels, each element of a
finite group appears exactly once in
each row of the Cayley table for the
group.
Theorem.
Ignoring column labels, each element of a
finite group appears exactly once in each
column of the Cayley table for the group.
Definition.
Define integral powers of a∈G:
a0=ea1=aa2=aa⋮an+1=ana
so that an equals the product
of n
a's
for each positive integer n.
Also, for each positive integer n,
a−n=(a−1)n
Theorem.
Laws of exponents.
For all integers m,n,
aman=am+n(am)n=amn
Note.
In additive notation, for all integers m,n:
na=a+⋯+a(n terms)(−n)a=n(−a)(ma)+(na)=(m+n)an(ma)=(mn)a
Definition.
If G is a group and a∈G
then ⟨a⟩
will denote the set of all integral
powers of a.
⟨a⟩={an∣n∈Z}
Example 14.1.
The set of all integral powers of
(1 2 3) is
⟨(1 2 3)⟩={(1),(1 2 3),(1 3 2)}
Theorem 14.2.
If G is a group and a∈G
then ⟨a⟩,
the set of all integral powers of a,
is a subgroup of G.
The subgroup ⟨a⟩,
is called
the subgroup generated by a.
Definition.
If H is a subgroup and
H=⟨a⟩,
for some a∈H,
then H is said to be a
cyclic subgroup.
Guevara Corollary.
If a∈G, then the subgroup
⟨a⟩ is cyclic.
Example.
The group of integers is cyclic:
Z=⟨1⟩=⟨−1⟩
Theorem 14.3.
If G is a group, a∈G,
and r,s∈Z such that r≠s
and ar=as, then
-
There is a smallest positive integer n
such that an=e.
-
If t is an integer, then at=e
iff n is a divisor of t.
-
The elements
e=a0,
a,
a2,
…,
an−1
are distinct, and
⟨a⟩={e,a,a2,…,an−1}.
Definition.
If a is an element of a group,
then the smallest positive integer n
such that an=e, if it exists,
is called
the order of a.
If there is no such integer, then a
is said to have
infinite order.
The order of an element a will be
denoted by o(a).
Note.
If a∈G then an=e
automatically if n=0.
By the preceding definition,
the order of a is never 0
That is, if n is the order
of a, then n>0 by definition,
and is the smallest such n
that an=e.
Example 14.2.
-
In S3, o((1 2 3))=3.
-
In the group of nonzero rational numbers
(operation multiplication),
2 has infinite order,
because 2n≠1 for every
positive integer n.
Note.
In additive notation, the condition
an=e becomes na=0.
In Zn, the condition an=e
becomes n[a]=[0].
Example 14.3.
In Z6, o([2])=3,
because [2]≠[0] and
2[2]=[2]⊕[2]=[4]≠0
but
3[2]=[2]⊕[2]⊕[2]=[6]=[0].
(We also see that ⟨[2]⟩={[0],[2],[4]}.)
Corollary.
If a is an element of a group,
then o(a)=|⟨a⟩|.