Summary
Highlights
A subgroup H of a group G is called a normal subgroup if its left and right cosets are equal for all elements A in G (aH = Ha). This is denoted by a special symbol like a subgroup symbol with an extra line. It's important to note that this is not saying aH must equal Ha for all 'a' and 'h' elements, but that the sets of left and right cosets are equal.
The video revisits an example from cosets where H = {1, 11} in U30. Here, 7H is shown to be equal to H7, demonstrating that H is a normal subgroup. A non-example is shown in D4 (symmetries of a square) where H = {e, r^2}. While HR and RH can be equal for some elements like r^2, they are not equal for all elements like r, proving it's not a normal subgroup. Another example in D4 with H = {e, r^2, RF, r^3F} demonstrates that even if aH does not strictly equal Ha for individual elements, the sets of left and right cosets can still be equal, making it a normal subgroup.
To prove a subgroup is normal, the 'Normal Subgroup Test' is introduced: A subgroup H is normal in G if and only if for all X in G, XHX⁻¹ (the conjugate) is a subset of H. This means that for every X in G and every h in H, XhX⁻¹ must be in H. This method is often easier than directly proving the equality of left and right cosets.
The normal subgroup test is applied to the two previous examples. For H = {1, 11} in U30, it is shown that all conjugates XHX⁻¹ result in the set {1, 11}, which is H itself, confirming it's a normal subgroup. For H = {e, RF} in D4, the conjugates produce elements like e, RF, and R³F. Since R³F is not in the original H, H is not a normal subgroup, aligning with the earlier observation.
The video briefly explains the underlying reason for the normal subgroup test's validity. It connects the concept of conjugation to the group operation, demonstrating how, through algebraic manipulation, the condition XhX⁻¹ belonging to H ensures the closure property required for normal subgroups in the context of coset multiplication.
Several key properties of normal subgroups are discussed: If G is abelian, any subgroup H of G is normal. The center of a group is always normal. The trivial subgroup (containing only the identity) and the group itself are always normal subgroups. Other properties mentioned for self-study include An being a normal subgroup of Sn, subgroups with an index of two, subgroups of Dn consisting solely of rotations, and all subgroups of cyclic groups being normal due to their abelian nature.
Two practice problems are presented. The first asks if H = {e, (1 2)} is normal in S3. By taking x = (1 3) and computing xhx⁻¹, it's found that (1 3)(1 2)(1 3)⁻¹ = (2 3), which is not in H, thus H is not normal in S3. The second problem asks if SL2(R) (2x2 matrices with determinant 1) is a normal subgroup of GL2(R) (all invertible 2x2 matrices). By using the property of determinants, det(XHX⁻¹) = det(X)det(H)det(X⁻¹) = det(X) * 1 * (1/det(X)) = 1, proving that XHX⁻¹ is always in H, and thus SL2(R) is a normal subgroup of GL2(R).