Normal Subgroups

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Normal Subgroups Definition A subgroup H of a group G is said to be normal in G, denoted H/G, if the left and right cosets gH and Hg are equal for all g 2 G. Equivalently, H is normal in G if gHg−1 = fghg−1 : h 2 Hg is a subset of H for every g 2 G. Example Recall that GL2(R) is the group of invertible, 2 × 2 matrices with real entries under matrix multiplication. The subgroup SL2(R) of 2 × 2 invertible matrices with determinant 1 (called the special linear group) is normal in GL2(R). To −1 see this, let g be any matrix in GL2(R), and let h be any matrix in SL2(R). We need to check that ghg is in SL2(R); that is, that it is also a matrix with determinant 1: −1 −1 −1 −1 −1 det(ghg ) = det(g) det(h) det(g ) = det(g)1 det(g ) = det(g) det(g ) = det(gg ) = det(I2) = 1 Finding normal subgroups using this definition can be difficult, as can proving that a subgroup is normal. Fortunately, we have several theorems that help with specific situations. Theorem 1 The trivial subgroup (containing only the identity) is normal in every group, and every group is a normal subgroup of itself. Theorem 2 Every subgroup of an Abelian group is normal. Theorem 3 The center of a group is normal in the group. Recall that the index of a subgroup H in a group G is the number of distinct left cosets of H in G. Theorem 4 Every subgroup of index 2 is normal. The previous theorem can be combined with Lagrange's theorem to quickly find a normal subgroup. By Lagrange's theorem, the index of a subgroup H in a group G is equal to jGj=jHj. So, if we find a subgroup H whose order is exactly half the order of G, then H will be normal. This leads to the following theorem: Theorem 5 The alternating group An is normal in Sn for any n. Example Not every subgroup of S3 is normal in S3. For an example, let H = f(1)(2)(3); (1 3)(2)g. If we take σ = (1 2)(3), −1 we see that σHσ is not contained in H, so H is not normal in S3. σHσ−1 = f(1 2)(3)(1)(2)(3)(1 2)(3); (1 2)(3)(1 3)(2)(1 2)(3)g = f(1)(2)(3); (1)(2 3)g Theorem 6 The kernel of a homomorphism (or isomorphism) φ : G1 ! G2 is always a normal subgroup of G1. Theorem 7 If H is a finite subgroup of G, and H is the only subgroup of G with order jHj, then H is normal in G. Math 280 Fall 2015 Part 2: Groups.

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