Page 65 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 2. Another direct proof: If a is coprime to n, then multiplication by a permutes the residue
classes mod n that are co prime to n; in other words, (writing R for the set consisting of the
(n) different such classes) the sets { x : x in R } and { ax : x in R } are equal; therefore, the two
products over all of the elements in each set are equal. Hence, P  a P (mod n) where P is
(n)
the product over all of the elements in the first set. Since P is coprime to n, it follows that
a  1 (mod n).
(n)
Self Assessment

1. If H is a subgroup of a group G, and let x  G then Hx = {Hx | h  H}. Thus Hx called as
(a) Left coset of H is G (b) Right coset of H is G
(c) Subgroup of G (d) Cyclic group of G

2. If H is .................., the alternating group on 3 symbols
(a) A1 (b) A3
(c) A4 (d) A5
3. Every group of prime order is ..................
(a) normal (b) cyclic

(c) subgroup (d) abelian
4. Two left cosets of a .................. are disjoint or identical
(a) normal (b) cyclic

(c) subgroup (d) abelian
5. a = 1 (modn) where a, n  N, (a, n) = 1 and
f(n)
(a) n  2 (b) n  2
(c) n = 2 (d) n  2

4.3 Summary

The definition and examples of right and left cosets of a subgroup.

Two left (right) cosets of a subgroup are disjoint or identical.

Any subgroup partitions a group into disjoint left (or right) cosets of the subgroup.

The definition of the order of a group and the order of an element of a group.

The proof of Lagrange’s theorem, which slates that if H is a subgroup of a finite group G,

then o(G) = o(H) | G : H |. But, if m | o(G), then G need not have a subgroup of order.
The following consequences of Lagrange’s theorem:

Every group of prime order is cyclic.

a = 1 (mod n), where a, n  N, (a,n) = 1 and n  2.
(n)


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