Some examples of groups and subgroups
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This article is a very elementary introduction to some typical examples of groups and subgroups. The idea is to illustrate basic principles and ideas of group theory.
Groups that we have all seen
Integers, rationals and reals
- The integers under addition form a group. The identity element of the group is , and the additive inverse is just the usual negative. In fact, the group of integers is an Abelian group: addition is commutative for integers.
- The rational numbers under addition form a group. The identity element of the group is , and the additive inverse is just the usual negative. This group is Abelian, and the integers form a subgroup.
- The real numbers also form a group under addition. The rational numbers form a subgroup of the group of real numbers, and the integers form a smaller subgroup.
- The nonzero rational numbers under multiplication form a group. The identity element for this group is . This group is also Abelian.
More generally, given any field, the field is a group under addition, and the nonzero elements of the field form a group under multiplication.
Some non-examples of groups are:
- The natural numbers under addition: There is no additive identity and there are no additive inverses.
- The nonzero integers under multiplication: The nonzero integers under multiplication have a multiplicative identity (namely ). Hence, they form a monoid. But not every nonzero integer has an integer as its multiplicative inverse. In fact, the only invertible elements are .
Modular arithmetic: groups from number theory
Further information: cyclic group,cyclic implies Abelian One of the ways of constructing finite groups is to look at integers modulo a given nonzero integer . By integers modulo , we mean that we are looking at the group of integers, modulo the equivalence relation of differing by a multiple of . For instance, modulo 2, there are exactly two equivalence classes of numbers: the even numbers and odd numbers. Thus, the group of integers modulo 2, termed the cyclic group of order two, has exactly two elements, one corresponding to the collection of even numbers and one corresponding to the collection of odd numbers. These are typically represented as and . The group operation is then given by:
Similarly, modulo 4, there are four equivalence classes of numbers: the multiples of 4, the numbers that leave a remainder of 1 modulo 4, the numbers that leave a remainder of 2 modulo 4, and the numbers that leave a remainder of 3 modulo 4.
The equivalence classes of numbers modulo form a group under addition. For instance, whenever we add a number that is mod and a number that is mod , we get a number that is mod .
For convenience, we represent an equivalence class modulo by the smallest nonnegative integer representative. So the four equivalence classes modulo 4 are represented by the elements respectively, and while adding, we reduce the sum modulo 4 (so ).
Groups that are obtained in this way are termed cyclic groups.
Another way of viewing cyclic groups is as quotients of the group of integers by a normal subgroup.
Permutations: groups from functions
Further information: symmetric group
A permutation of a set is a bijective map from to itself. The symmetric group on a set is the set of all permutations on it, where:
- The product of two permutations is composition. If and are permutations, their product is the map
- The identity map is the identity element
- The inverse of a permutation is its inverse as a function
The symmetric groups are important examples of non-Abelian groups: in fact the symmetric group on a set of size at least three, is always non-Abelian. Moreover, it is surprisingly true that every finite group occurs as a subgroup of the symmetric group, so symmetric groups are subgroup-rich. Further information: Cayley's theorem
Transformations: groups from geometry
Further information: Groups as symmetry
Yet another example of groups comes from geometry. In the geometric context, groups occur as symmetries of structure that preserve certain geometric properties. As in the case of the groups of permutations, the multiplication is by composition, the identity element is the identity map, and the inverse map sends a transformation to its inverse function.
For instance, the rotations of the plane about a point form a group (in fact, an Abelian group). Here, each rotation is described by a (signed) angle of rotation, and to add two rotations , we add their angles. However, the angles are viewed modulo , so the group of rotations is the group of real numbers quotiented by the equivalence relation of differing by .
Non-Abelian groups also arise in geometry. In fact, rotations about a fixed point in three-dimensional space form a non-Abelian group. Rotations around different axes do not, in general, commute.
Constructing groups with certain subgroup structures
One question of interest is: can we find groups with certain structural attributes? Questions about existence of groups are usually very complicated, and we consider here a very simple version of these questions.
Every nontrivial group has at least two subgroups: the trivial subgroup and the group itself. What are the groups for which there are no other subgroups?
To solve this, we use the following ideas:
- Given any nontrivial element of the group, we can consider the cyclic subgroup generated by that. This is the smallest subgroup containing that element.
- If the group has no nontrivial proper subgroup, then the cyclic subgroup generated by that element must be the whole group. In particular, the group must be cyclic.
- So the group must either be the group of integers, or it must be the group of integers modulo . But the group of integers has lots of subgroups (for instance, multiples of ). The group of integers modulo has proper nontrivial subgroups if is not prime.
- So the only possibility for a group with no proper nontrivial subgroup is a cyclic group of prime order
- Conversely, any cyclic group of prime order has no proper nontrivial subgroup