Let G be a group and let g ? G. Show that g commutes with each of its conjugates in G if and only if g belongs to an abelian normal subgroup of G.
Added by Alvaro L.
Close
Step 1
We want to show that g belongs to an abelian normal subgroup of G. Consider the set H = {xgx^-1 : x € G}. This is the conjugacy class of g in G. Since g commutes with each of its conjugates, we have xgx^-1 = gx^-1x for all x € G. This implies that H is a Show more…
Show all steps
Your feedback will help us improve your experience
Madhur L and 51 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
) Let N be a normal subgroup of the group G. Prove that the quotient group G/N is abelian if and only if N contains the commutator subgroup [G, G].
Madhur L.
Suppose that G is a finite abelian group and has exactly one subgroup for each divisor of |G|. Show that G is cyclic.
Sri K.
Let G be a group of order 203. Prove that if G has a normal subgroup of order 7 then G is abelian.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD