F igure F is a regular hexagon with an equilateral triangle inside it. The verti

ID: 2967686 • Letter: F

Question

Figure F is a regular hexagon with an equilateral triangle inside it. The vertices are labelled 1 to 6.

Write down in two-line form the elements of the symmetry group S(F). Describe each symmetry geometrically. Identify each of these symmetries by a single letter, and use these letters to construct a group table for S(F).

The group S(F) is isomorphic to one of the following groups:

(Z6, +6), S(triangle), S(Hexagon). Determine to which of these groups S(F) is isomorphic, justifying your answer, please.

Write down one subgroup of S(F) of order 2, and one of order 3, please.!!

Explanation / Answer

F is an object which is a one-dimensional regular hexagon with an equilateral triangle inscribed, where the triangle and hexagon cannot move independently.

The isometries of F (making up the symmetry group of F, S(F) ) are as follows - 6 rotational positions of the object, and 2 sides from which to view the object (back or front), making 12 isometries altogether. This is the same as the symmetric group of the hexagon S(Hexagon) assuming that the inscribed triangle is fixed and cannot be moved independently of the hexagon it is inscribed on. S(Hexagon) is also known as the dihedral group of order 6, D6 (dihedral objects are polyhedral shapes viewable from both sides - back or front).

Looking at the attached figure, the symmetries/isometries/group elements are labelled as follows:

a - identity

b,c,d,e,f - rotations of 120,240,60,180,300 degrees respectively

g - reflection about the left diagonal

h,j - rotations of 120,240 degrees respectively, each followed by a reflection about the left diagonal of the hexagon

k,l,m - rotations of 60,180,300 degrees respectively, each followed by a reflection about the right diagonal of the hexagon.

Two-line form (where a= (123456) is taken as the first line and all other lines other the relevant second line of the two-line form):

`((underline(a)|,underline(123456)),(b|,561234),(c|,345612),(d|,612345),(e|,456123),(f|,234561),(g|,624351),(h|,432165),(j|,216543),(k|,165432),(l|,543126),(m|,321654)) `

Group table of elements of S(F) (1st operation = rows, 2nd operation = columns)` ` :

Notice that a,e,g,h,j,k,l,m are their own inverses

The symmetry group of F, S(F), is the same as (isomorphic to) the symmetry group of the regular hexagon S(Hexagon) which is also the dihedral group D6. If the inscribed triangle could move independently from the hexagon it is inscribed on, there would be more isometries to the group. However, it is assumed that the triangle and hexagon are a fixed single body, so that it would behave in its symmetries in precisely the same way as a hexagon with no inscribed triangle. The triangle then just makes the particular orientation or isometry of the hexagon more identifiable. Therefore S(F) is equivalent to S(Hexagon).

A subgroup of S(F) of order 2 is for example (a,g), that is, the subgroup consisting of the identity element and the reflection about the left diagonal. We have a o a = a, a o g = g, g o a = g and g o g = a, since the operation g is its own inverse.

A subgroup of S(F) of order 3 is for example (a,b,c) - the subgroup consisting of the identity element, a rotation about the centre of 120 degrees, and a rotation about the centre of 240 degrees. We have a o a = a, a o b = b, a o c = c, b o a = b, c o a = c, b o c = a, c o b = a since b and c are the inverse operation of each other.

Hope this helps:-)

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