Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 19G / i

Table of contents
- Solution i

Solution

Every group element has a unique form

a^{r} b^{s}

, with

r mod 7

and

s mod 3

. Conjugation by

b

doubles the exponent of

a

, giving the two orbits

{1, 2, 4}

and

{3, 6, 5}

. Conjugation by powers of

a

runs through all seven elements of each nonidentity coset of

⟨ a ⟩

. The classes are therefore

{1}, {a, a^{2}, a^{4}}, {a^{3}, a^{5}, a^{6}}, {a^{r} b : 0 \leq r < 7}, {a^{r} b^{2} : 0 \leq r < 7} .

Their sizes

1, 3, 3, 7, 7

sum to twenty-one.

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