1 Let 4ABC denote an equilateral triangle with vertices A B

1. Let 4ABC denote an equilateral triangle with vertices A, B, and C. Let D3 denote the group of symmetries of the triangle. Verify D3 can be generated by dB (reflection about the bisector of angle ?ABC) and R120 (rotation by 120? ).

Solution

there are 6 motoion

1) Do nothing or rotate by 360 it is denoted by e,

2) rotate by 120 counter clock wise is denoted by a,

3) rotate 240 deg counterclock wise is denoted by b,

4) x stands for the flip about the axis through the top vertex.( reflection about the bisector of the top angel)

5) y stands for the flip about the axis through the lower left vertex.

6) z stands for the flip about the axis through the right lower vertex.

now do nothing is the identity element,

2) rotate 120 deg is inverse of rotate 24o,

and the each flip are its inverse .

now we are forming caley table

now from the caley table it can be proved that a, x together can generate the group

since a.a=b

,a.x=y,

now a.a.a=b.a=e,

and a^4= a^3.a=a

a^2.x=b.x=z, and b^3=b^2.b=b

hence the rotation 120(a) and the reflection about bisector of any angel can generate the group.