Alternating group
The alternating group
Simplicity
An important property of the alternating group
for is generated by 3-cycles. - If
with contains a 3-cycle, then . - Every nontrivial
for contains a 3-cycle.
Proof
Since pairs of transpositions generate
for by construction, we need only show that any pair of transpositions can be written as a product of 3-cycles. Noting that , the following list exhausts any pair of transpositions: proving ^S1.
Now
for can in fact be generated only from 3-cycles of the form with fixed in , since every 3-cycle can be expressed as such: Now assume
contains a 3-cycle, say . By normality it follows for any Hence
contains all 3-cycles of the form and hence all 3-cycles, thus by ^S1 it is , proving ^S2. Now let
for be nontrivial. Then one of the following holds: Case 1: Suppose there exists
with a cycle of length , so without loss of generality (by relabelling) where and are disjoint. Since , it follows so
contains a 3-cycle. Case 2a: Suppose there exists a
which contains two disjoint 3-cycles (and nothing longer). Without loss of generality, for disjoint , , and . Since , it follows which falls under case 1.
Case 2b: Suppose there exists a
containing exactly one 3-cycle and otherwise only transpositions. Without loss of generality where and are disjoint, and . Then , so contains a 3-cycle. Case 2c: If
contains a 3-cycle we are already done. Case 3: The only remaining possibility is that there exists a
which is a product of disjoint transpositions, and an even number thereof since . Without loss of generality with , , disjoint and . Then whence
thus
contains a 3-cycle. This proves ^S3 and therewith the simplicity of
for .
Note that
Properties
is -transitive.