Alternating group
The alternating group \(A_n\) is defined as a certain subgroup of the symmetric group \(S_n\): namely, the collection of all elements which can be made by multiplying together an even number of transpositions. This is a well-defined notion (proof).
Examples
A cycle of even length is an odd permutation in the sense that it can only be made by multiplying an odd number of transpositions. For example, \((132)\) is equal to \((13)(23)\).
A cycle of odd length is an even permutation, in that it can only be made by multiplying an even number of transpositions. For example, \((1354)\) is equal to \((54)(34)(14)\).
The alternating group \(A_4\) consists precisely of twelve elements: the identity, \((12)(34)\), \((13)(24)\), \((14)(23)\), \((123)\), \((124)\), \((134)\), \((234)\), \((132)\), \((143)\), \((142)\), \((243)\).
Properties
\(A_n\) is generated by its \(3\)-cycles. (Proof.)
The conjugacy classes of \(A_n\) are easily characterised.
Children:
- The collection of even-signed permutations is a group
This proves the well-definedness of one particular definition of the alternating group.
- Alternating group is generated by its three-cycles
A useful result which lets us prove things about the alternating group more easily.
- The alternating groups on more than four letters are simple
The alternating groups are the most accessible examples of simple groups, and this fact also tells us that the symmetric groups are “complicated” in some sense.
- The alternating group on five elements is simple
The smallest (nontrivial) simple group is the alternating group on five elements.
- Conjugacy classes of the alternating group on five elements
\(A_5\) has easily-characterised conjugacy classes, based on a rather surprising theorem about when conjugacy classes in the symmetric group split.
- Splitting conjugacy classes in alternating group
The conjugacy classes in the alternating group are usually the same as those in the symmetric group; there is a surprisingly simple condition for when this does not hold.
Parents:
- Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.