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 $0$, 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 $0$, 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 $1$. 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 $1$). Hence, they form a monoid. But not every nonzero integer has an integer as its multiplicative inverse.

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 $n$. By integers modulo $n$, we mean that we are looking at the group of integers, modulo the equivalence relation of differing by a multiple of $n$. For instance, modulo 2, there are exactly two equivalence classes of numbers: the even numbers and odd numbers. 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 $n$ form a group under addition. For instance, whenever we add a number that is $1$ mod $4$ and a number that is $2$ mod $4$, we get a number that is $3$ mod $4$.

For convenience, we represent an equivalence class modulo $n$ by the smallest nonnegative integer representative. So the four equivalence classes modulo 4 are represented by the elements $0,1,2,3$ respectively, and while adding, we reduce the sum modulo 4 (so $2+3=1$).

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 $S$ is a bijective map from $S$ 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 $f$ and $g$ are permutations, their product is the map $x\mapsto f(g(x))$
• 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.

Transformations: groups from geometry

Further information: Groups as symmetry