Group Theory III.1 (Subgroups: Definition)

The astute reader might have noticed in our list of examples of groups that some groups are contained in others. E.g. A_n \subset S_n, SL_n(\mathbf{R}) \subset GL_n(\mathbf{R}), the group of symmetries of the icosahedron is contained in S12 etc. Furthermore, this subset inherits the group operation from the big group.

Warning : be careful with the cyclic groups Cn. For example, even though we have C_5 \subset C_{10} set-theoretically (recall that the former is {0, …, 4} while the latter is {0, …, 9}), the group operation is inconsistent. E.g. 3 + 4 = 2 in C5 but 3 + 4 = 7 in C10. Nevertheless, it is possible to “embed” C5 into C10, as we shall see later.

Just as sets have subsets, it is only natural to define the following: a subgroup of G is a subset H which forms a group under the * operation inherited from G. To ensure that (H, *) forms a group as well, we need to go back to check those 3 axioms of a group. Associativity clearly comes for free, and the only remaining conditions to check are:

  • e \in H;
  • if h, h' \in H, then the product hh' \in H;
  • if h \in H, then the inverse h^{-1} \in H.

In fact, it suffices to reduce to the following conditions.

A subset H of G is a subgroup if and only if H is non-empty, and for any h, h' \in H, we have h h'^{-1} \in H.

Clearly the boxed condition is necessary for H to be a subgroup. The proof for sufficiency is easy and left as an exercise. Note that non-emptiness of H is important, since without it, the empty set would also become a subgroup of G, whereas our earlier definition forces H to contain e at least. This reason may appear cosmetic, but disallowing empty sets as subgroups turns out to be wise in hindsight when we encounter Lagrange’s theorem in future.

 


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