Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 3 / 5D (Algebra and Geometry)

Use $D_{12}$ for the dihedral group of order twelve and write each element uniquely as $g^j$ or $g^{j}h$, $0\le j<6$. The classification of subgroups of a dihedral group organizes the complete list: rotational subgroups are cyclic, and a subgroup containing a reflection is generated by its rotational part and one reflection. The individual lists and quotients follow below.

The involutions in an even dihedral group are the half-turn $g^3$ and the six reflections $g^{j}h$. Thus the seven order-two subgroups areThe half-turn is central, so only $⟨g^3⟩$ is normal. Indeed $g(g^{j}h)g^{-1}=g^{j+2}h$, a different reflection; an order-two subgroup has only one nonidentity element, so each reflection subgroup fails the normal subgroup condition.

The remaining proper subgroups areTheir orders are respectively one, three, six, four and six. The order-four groups are Klein four-groups; the last two are dihedral groups of order six, isomorphic to $S_3$.

For completeness, if $H$ contains a reflection, write $H\cap ⟨g⟩=⟨g^d⟩$ with $d\mid 6$. Any other reflection differs from a chosen $g^{j}h$ by an element of this rotational subgroup, so $H=⟨g^d,g^{j}h⟩$ with $j$ specified modulo $d$. The choices $d=1,2,3,6$ give respectively the whole group, the two order-six groups, the three order-four groups and the six reflection groups already listed. This proves there are no omitted subgroups.

Thus there are eight remaining proper subgroups. The printed count nine includes one too many; nine would be correct if the whole group were included with these remaining subgroups. Altogether $G$ has sixteen subgroups, fifteen proper.

The proper normal subgroups and their quotient groups are as follows. Here the trivial subgroup is included as a proper subgroup.

The rotational subgroups are preserved by reflection conjugation, and the last three order-six subgroups are normal by index two. The three order-four groups are not normal, since conjugation by $g$ changes the reflection exponent by two, which is nonzero modulo three. Part solution excluded the reflection subgroups. These observations, together with the complete subgroup list, prove the normal-subgroup list is exhaustive.

The rotation $g$ has order six. The possible cycle types in the alternating group $A_4$ are the identity, a three-cycle, and a product of two disjoint transpositions; their orders are one, three and two. Since a group isomorphism preserves the order of a group element, $D_{12}$ is not isomorphic to $A_4$.