Question

50. Prove that if $U$ and $W$ are normal subsets of a Sylow $p$-subgroup $P$ of $G$ then $U$ is conjugate to $W$ in $G$ if and only if $U$ is conjugate to $W$ in $N_G(P)$. Deduce that two elements in the center of $P$ are conjugate in $G$ if and only if they are conjugate in $N_G(P)$. (A subset $U$ of $P$ is normal in $P$ if $N_P(U) = P$.)

Name: 50 prove that if u and w are normal subsets of a sylow p subgroup p of g then u is conjugate to w in g if and only if u is conjugate to w in ngp deduce that two elements in the center of p 73219
Uploaded: 2024-06-21T21:21:20-08:00
Duration: 11 s
Channel: Armani Willis
Description: 50 prove that if u and w are normal subsets of a sylow p subgroup p of g then u is conjugate to w in g if and only if u is conjugate to w in ngp deduce that two elements in the center of p 73219

          50. Prove that if $U$ and $W$ are normal subsets of a Sylow $p$-subgroup $P$ of $G$ then $U$ is conjugate to $W$ in $G$ if and only if $U$ is conjugate to $W$ in $N_G(P)$. Deduce that two elements in the center of $P$ are conjugate in $G$ if and only if they are conjugate in $N_G(P)$. (A subset $U$ of $P$ is normal in $P$ if $N_P(U) = P$.)

50 prove that if u and w are normal subsets of a sylow p subgroup p of g then u is conjugate to w in g if and only if u is conjugate to w in ngp deduce that two elements in the center of p 73219

Added by Christine J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Armani Willis

06/21/2024

Step 1

Suppose $U$ is conjugate to $W$ in $G$. Then there exists $g \in G$ such that $gUg^{-1} = W$. If $g \in N_G(P)$, then $U$ is conjugate to $W$ in $N_G(P)$. Suppose $U$ is conjugate to $W$ in $N_G(P)$. Then there exists $g \in N_G(P)$ such that $gUg^{-1} = W$. Since Show more…

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50. Prove that if $U$ and $W$ are normal subsets of a Sylow $p$-subgroup $P$ of $G$ then $U$ is conjugate to $W$ in $G$ if and only if $U$ is conjugate to $W$ in $N_G(P)$. Deduce that two elements in the center of $P$ are conjugate in $G$ if and only if they are conjugate in $N_G(P)$. (A subset $U$ of $P$ is normal in $P$ if $N_P(U) = P$.)