Dihedral group

In group theory, a dihedral group is a group whose elements correspond to a closed set of rotations and reflections in the plane. The dihedral group with 2n elements is usually written as D2n. It is generated by a single rotation r with order n, and a reflection f with order 2.

(Note that some authors use the notation Dn instead of Wikipedia's notation D2n.)

The simplest dihedral group is D4, which is generated by a rotation r of 180 degrees, and a reflection f across the y-axis. The elements of D4 can then be represented as {e, r, f, rf}, where e is the identity or null transformation.

D4 is isomorphic to the Klein four-group.

If the order of D2n is greater than 4, the operations of rotation and reflection in general do not commute and D2n is not abelian; for example, in D8, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees.

Whatever the order of the dihedral group, the rotation r and the reflection f always satisfy

r f = f r -1.

Thus, beyond their obvious application to problems of symmetry in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups.

Some equivalent definitions of D2n are:

  • The symmetry group of a regular polygon with n sides (if n ≥ 3).
  • The automorphism group of the graph consisting only of a cycle with n vertices (if n ≥ 3).
  • The group with presentation ({r,f}; {rn, f ², (rf)²}).
  • The semidirect product of cyclic groups Cn and C2, with C2 acting on Cn by inversion (thus, D2n always has a normal subgroup isomorphic to Cn)

The number of subgroups of D2n (n ≥ 3), is equal to d(n) + σ(n), where d(n) is the number of positive divisors of n and σ(n) is the sum of the positive divisors of n.

In addition to the finite dihedral groups, there is the infinite dihedral group D. Every dihedral group is generated by a rotation r and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer n such that rn will be the identity. If the rotation is not a rational multiple, then there is no such n; the resulting group is then called D. It has presentation ({a,b}; {a², b²}}, and is isomorphic to a semidirect product of Z and C2.

D can also be visualized as the automorphism group of the graph consisting of a path infinite to both sides.

Finally, if H is any non-trivial finite abelian group, we can speak of the generalized dihedral group of H (sometimes written Dih(H)). This group is a semidirect product of H and C2, with order 2*order(H), a normal subgroup of index 2 isomorphic to H, and having an element f such that, for all x in H,  f -1 x f = x -1.


">
" size=20>

 
 

Browse articles alphabetically:
#0">0 | #1">1 | #2">2 | #3">3 | #4">4 | #5">5 | #6">6 | #7">7 | #8">8 | #9">9 | #_">_ | #A">A | #B">B | #C">C | #D">D | #E">E | #F">F | #G">G | #H">H | #I">I | #J">J | #K">K | #L">L | #M">M | #N">N | #O">O | #P">P | #Q">Q | #R">R | #S">S | #T">T | #U">U | #V">V | #W">W | #X">X | #Y">Y | #Z">Z