Sylow and TI implies CDIN

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 TI-subgroup) must also satisfy the second subgroup property (i.e., CDIN-subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about Sylow TI-subgroup|Get more facts about CDIN-subgroup

Statement

Suppose $G$ is a finite group, $p$ is a prime, and $P$ is a $p$ -Sylow subgroup of $G$ that is also a TI-subgroup (?): any conjugate of $P$ is either equal to $P$ or intersects it trivially. Then $P$ is a CDIN-subgroup of $G$ : any two elements of $P$ that are conjugate in $G$ are conjugate in $N_{G} (P)$ .

In fact, something stronger is true: the original conjugating element is itself in $N_{G} (P)$ .

Related facts

Facts used

Proof

CONVENTION WARNING: This article/section uses the right-action convention. The left and right action conventions are equally powerful and statements/reasoning here can be converted to the alternative convention (the main reason being that every group is naturally isomorphic to its opposite group via the inverse map). For more on the action conventions and switching between them, refer to switching between the left and right action conventions.

Given: A finite group $G$ , a prime $p$ , a $p$ -Sylow subgroup $P$ of $G$ such that the intersection of $P$ with any distinct conjugate of itself is trivial.

To prove: If $x, y \in P$ are conjugate by $g \in G$ , then $x$ and $y$ are conjugate in $N_{G} (P)$ . In fact, $g \in N_{G} (P)$ .

Proof: If $x$ or $y$ is the identity element, so is the other, and in this case we are done. So, we can assume that neither $x$ nor $y$ is the identity element.

By fact (1), there exist $p$ -Sylow subgroups $Q_{1}, Q_{2}, \dots, Q_{n}$ such that $P \cap Q_{i}$ is a tame intersection, and elements $g_{i} \in N_{G} (P \cap Q_{i})$ such that $g = g_{1} g_{2} \dots g_{n}$ and $x^{g_{1} g_{2} \dots g_{r}} \in P \cap Q_{r + 1}$ for $1 \leq r \leq n - 1$ .

However, since $P$ is TI, its intersection with any other Sylow subgroup (note also fact (2)) is trivial. Since $x^{g_{1} g_{2} \dots g_{r}} \in P \cap Q_{r + 1}$ , we are forced to have that all the $P \cap Q_{i}$ are nontrivial, and hence, equal to $P$ . This forces $Q_{i} = P$ for all $i$ , hence $g_{i} \in N_{G} (P)$ . Hence, $g$ , which is the product of the $g_{i}$ s, is in $N_{G} (P)$ , and $x$ and $y$ are conjugate via an element in $N_{G} (P)$ .

References

Textbook references

Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 254, Theorem 4.6, Chapter 7 (Fusion, transfer and p-factor groups), ^{More info}