222 CHAPTER 4 Abstract Algebra
subtle than it looks, and shall be the topic of the next subsection.
A group (G,∗) is calledAbelian^10 if the operation∗is commutative.
Granted, it would make good sense to call such groups “commutative,”
but we enjoy naming concepts after influencial mathematicians. In the
above list of groups you should be able to separate the Abelian groups
from the non-Abelian ones.
Exercises
- Consider the two graphs given at the beginning of this section;
here they are again:
Write down the elements of the corresponding automorphism groups,
and then give the corresponding Cayley tables.- In the group (Z 17 ,·), find 2−^1 and 5−^1. Find any elementsxsuch
thatx^2 = 1. - LetX={ 1 , 2 , 3 }and consider the group Sym(X) of permutations
onX. Define the following two permutations:
σ=
1 2 3
↓ ↓ ↓
2 3 1
τ =
1 2 3
↓ ↓ ↓
1 3 2
(^10) after the Norwegian mathematics Niels Henrik Abel (1802–1829)