《Discrete Mathematics II教学-华南理工》Lecture 5 Subgroups.pdfVIP

I.5 Subgroups 1 Section I.5. Subgroups Note. Just as a vector space can have a subspace, as you see in linear algebra, a group can have a subgroup. As with vector spaces, this would be a subset with all the structure of a group. Definition 5.3. If G is a group, then the order |G| of G is the number of elements in G. Note. We will primarily study finite groups where the above definition is clear. However, we will occasionally study infinite groups (such as ⟨R, +⟩). I.5 Subgroups 2 Definition 5.4. If a subset H of a group G is closed under the binary operation of G and if H with the induced operation from G is itself a group, then H is a subgroup of G. We denote this as H ≤ G or G ≥ H . If H is a subgroup of G and H ≠ G, we write H G or G H . Example. ⟨Z, +⟩ ⟨Q, +⟩ ⟨R, +⟩ ⟨C, +⟩. ˆ ˆ ˆ ˆ ˆ Example From Linear Algebra. ⟨span{i, j }, +⟩ ⟨span{i, j, k}, +⟩. ∗ Example 5.8. The nth roots of unity in C form a subgroup Un of the group ⟨C , ·⟩ (recall that C∗ = C \ {0}). Definition 5.5. If G is a group, then G itself is a subgroup of G called the improper subgroup of G; all other subgroups are proper subgroups. The subgroup {e} is the trivial subgroup; all other subgroups are nontrivial subgroups. I.5 Subgroups 3 Example 5.9. There are two (nonisomorphic) groups of order 4. One is ⟨Z , + ⟩ = 4 4 Z : 4 +4 0 1 2