SECTION 4.2 Basics of Group Theory 235
- LetGbe a group, letHbe a subgroup, and recall the equivalence
relation ( modH) defined by
g≡g′( modH) ⇔g−^1 g′∈H.The equivalence classes inG relative to this equivalence relation
are called the (left)cosets of H in G. Are the cosets also sub-
groups ofG? Why or why not?- LetGbe the group of Exercise 3 and letKbe the cyclic subgroup
generated byστ. Compute the left cosets ofK inG. - Let G be the group of Exercise 3 and let L be the subgroup
{e, τ, σ^2 , σ^2 τ}. Compute the left cosets ofLinG. - Here we shall give yet another proof of the infinitude of primes.
Define, for each primepthe correspondingMersenne numberby
settingMp= 2p−1 (these are often primes themselves). Assume
by contradiction that there are only finitely many primes and let
pbe thelargestprime. Letq be a prime divisor ofMp= 2p−1.
Then we have, in the multiplicative groupZ∗q of nonzero integers
moduloq, that 2p≡1(modq). This says, by exercise 2 on page 227
thatpis the order of 2 in the groupZ∗q. Apply Lagrange’s theorem
to obtainp|(q−1), proving in particular thatq is a larger prime
thanp, a contradiction.
4.2.8 Homomorphisms and isomorphisms
What is the difference between the additive group (Z 6 ,+) and the mul-
tiplicative group (Z∗ 7 ,·)? After all, they are both cyclic: (Z 6 ,+) has
generator 1 (actially, [1]), and (Z∗ 7 ,·) has generater 3 ([3]). So wouldn’t
it be more sensible to regard these two groups as algebraically the same,
the only differences being purely cosmetic? Indeed, doesn’t any cyclic
group of order 6 look like{e, x, x^2 , x^3 , x^4 , x^5 , x^6 }?