Suppose that a group G has a subgroup of order n. Prove that the intersection of all subgroups of order n is a normal subgroup of G.
Added by Mary S.
Step 1
Let $H_1, H_2, \dots, H_k$ be all the subgroups of $G$ with order $n$. We want to show that their intersection, denoted by $H = \bigcap_{i=1}^k H_i$, is a normal subgroup of $G$. Show more…
Show all steps
Close
Your feedback will help us improve your experience
Madhur L and 61 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.
Let G be a group with N is a normal subgroup of G. Assume [x,y]N = [xN,yN] and N is a subgroup of [G,G]. Prove that [G,G]/N = [G/N, G/N]
Vincenzo Z.
Suppose G is a finite group of order mn and H is a normal subgroup of order m. Prove that, for any element g ∈ G, g^n ∈ H.
Sam S.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD