Sylow implies MWNSCDIN

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., Sylow subgroup) must also satisfy the second subgroup property (i.e., MWNSCDIN-subgroup)
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Statement

Any Sylow subgroup of a group is a MWNSCDIN-subgroup.

Definitions used

Sylow subgroup

Further information: Sylow subgroup

A subgroup $P$ of a group $G$ is termed a Sylow subgroup if its order is a power of a prime $p$ and the index of $P$ in $G$ is relatively prime to $p$.

MWNSCDIN-subgroup

Further information: MWNSCDIN-subgroup

A subgroup $H$ of a group $G$ is termed a MWNSCDIN-subgroup if, given a collection of normal subsets $A_i,i\in I$ and $B_i,i\in I$ of $H$, such that there exists $g\in G$ such that $g{A}_i{g}^{-1}=B_i$, for all $i\in I$, then there exists a $h\in N_G(H)$ such that $h{A}_i{h}^{-1}=B_i$ for each $i\in I$.

Related facts

Similar facts

• Sylow implies WNSCDIN
• Pronormal implies MWNSCDIN
• Pronormal implies WNSCDIN
• Sylow and TI implies CDIN
• Abelian Sylow implies SCDIN

Opposite facts

• Sylow not implies CDIN

Facts used

1. Sylow implies pronormal
2. Pronormal implies MWNSCDIN

Proof

Proof using given facts

The proof follows from facts (1) and (2).

Hands-on proof

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