dihedral group properties
1 Properties of Dihedral Groups
Contemporary group theorists prefer over as the notation for the dihedral group of order . Although this notation is overly explicit, it does help to resolve the ambiguity with the Lie type which corresponds to the orthogonal group . However, introductory texts in algebra still make use of the more appropriate notation to emphasize the connection to the symmetries of a regular -gon (-sided polygon).
, are the two abelian examples of dihedral groups. They can be considered as dihedral groups of the respective order because they satisfy the relations, though the geometric interpretations are slightly modified. Often can be termed the symmetries of a line segment, and the the symmetries of a non-square rectangle, as the symmetry groups of each of these object is isomorphic to and respectively. These exceptions cause problems for most theorems on dihedral groups so it is convenient to insist that for theorems.
For the conjugacy classes note that so that these conjugacy classes are established. Next
for all . When we have so is invertible modulo . We let for any and we see that is conjugate to any . However, when we have a parity constraint that so far creates the two classes. We need to also verify conjugation by does not fuse the two classes. Indeed
thus we retain two conjugacy classes amongst the reflections.
Finally, the order of the elements is 2 – a fact used already. Thus the only cyclic subgroup of order , when , is and thus by its uniqueness it is characteristic. ∎
We will prove this with a more general claim. First we pause to note that as a corollary to these two propositions we can determine the entire lattice of normal and characteristic subgroups of a dihedral group.
If is normal and contains an element of the form , then it contains the entire conjugacy class of . If is odd then all reflections are conjugate to so contains all reflections of and so is as the relfections generate .
If instead is even then is forced only to contain one of the two conjugacy classes of reflections. If is even then contains and so it contains . If is odd then contains and so it contains (note as and ).
The two maximal subgroups of index which can exist when is even can be interchanged by an outer automorphism which maps and so these two are not characterisitic. The subgroups of a characterisitic cyclic group are necessarily characteristic. ∎
Quotient groups of dihedral groups are dihedral, and subgroups of dihedral groups are dihedral or cyclic.
For subgroups we proceed by induction. When the result is clear. Now suppose that has some proper subgroup that is not dihedral or cyclic. is contained in some maximal subgroup of . However the maximal subgroups of are cyclic or dihedral so falls to the induction step for – together with the fact that subgroups of cyclic groups are cyclic. Thus must actually be dihedral or cyclic to avoid contradictions. ∎
is nilpotent if and only if for some .
is solvable for all .
1.1 Automorphisms of
Let . The automorphism group of is isomorphic to , with the canonical action of . Explicitly,
with defined as
We apply the needle-in-the-haystack heuristic and search first to explain why these are the only possible forms for the automorphisms. We will then prove all such are indeed automorphisms.
Given , we know is characteristic in so for some . But is invertible so indeed so that . Next as cannot be sent to .
Now we claim .
Now we must show all are indeed homomorphisms when and . First we note that is well-defined as we have an irredundant listing of the elements. Next we verify the homomorphism cases.
So indeed is a homomorphism.
Finally, we show the composition of two such maps both to identify the automorphism group and to show that each is invertible.
|Title||dihedral group properties|
|Date of creation||2013-03-22 16:06:35|
|Last modified on||2013-03-22 16:06:35|
|Last modified by||Algeboy (12884)|