Let G be a group, let H < G, and let x E G. We use the notation xHx1 to denote the set of elements {xhx1h E H}. (xHx is called a conjugate of H.) Prove that xHx-1 is a subgroup of G (a) If H is finite, then how are |H| and |xHx (b) (c) related? Prove that H is isomorphic to xHx1
Let G be a group, let H < G, and let x E G. We use the notation xHx1 to denote the set of elements {xhx1h E H}. (xHx is called a conjugate of H.) Prove that xHx-1 is a subgroup of G (a) If H is finite, then how are |H| and |xHx (b) (c) related? Prove that H is isomorphic to xHx1
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 14E: Let H be a subgroup of a group G. Prove that gHg1 is a subgroup of G for any gG.We say that gHg1 is...
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