Subgroups and Group Properties in Abstract Algebra

Explore the concept of subgroups and their properties within group theory. This tutorial defines subgroups, examines criteria for subgroup identification, explores cyclic subgroups and cyclic groups, and provides examples to illustrate these fundamental concepts in abstract algebra.

Subgroups and Group Properties in Discrete Mathematics

Subgroups

A subgroup is a subset of a group that is itself a group under the same operation. Let (G, *) be a group, and H be a non-empty subset of G. H is a subgroup of G if it satisfies:

  1. Identity: The identity element of G is in H.
  2. Closure: If a and b are in H, then a * b is also in H.
  3. Inverses: If a is in H, then its inverse a-1 is also in H.

Cyclic Subgroups

A cyclic subgroup is a subgroup generated by a single element. Let (G, *) be a group, and x be an element of G. The cyclic subgroup generated by x is denoted as and contains all elements of the form xn where n is an integer.

Cyclic Groups

A cyclic group is a group that can be generated by a single element. If every element in a group G can be expressed as xn for some integer n, then G is a cyclic group.

Example: Cyclic Group

(The example from the original text showing the cyclic group G = {1, -1, i, -i} under multiplication, with i as the generator, should be included here.)

Abelian Groups

An abelian group is a group where the group operation is commutative—the order of the elements doesn't matter. Formally, a group (G, *) is abelian if a * b = b * a for all a, b ∈ G.

Example: Abelian Group

(The example from the original text proving that the non-zero real numbers under multiplication form an abelian group should be included here. Each group property—closure, associativity, identity, inverse, and commutativity—should be verified.)

Product of Groups

The direct product of two groups (G₁, *) and (G₂, *) is a new group (G₁ x G₂, *) formed by taking ordered pairs (g₁, g₂) where g₁ ∈ G₁ and g₂ ∈ G₂. The group operation is defined component-wise: (g₁', g₂') * (g₁'', g₂'') = (g₁' * g₁'', g₂' * g₂'').

(The proof that the direct product of two groups is also a group is given in the original text and should be included here.)

Cosets

Let H be a subgroup of a group G. A left coset of H in G is a set of the form xH = {xh | h ∈ H} for some x ∈ G. A right coset is a set of the form Hx = {hx | h ∈ H} for some x ∈ G. If the group operation is addition, these become x + H and H + x respectively.

Conclusion

Subgroups, cyclic groups, abelian groups, and the concept of cosets are fundamental concepts in group theory, providing a framework for understanding the structure and properties of groups.