Local powering-invariant subgroup containing the center is intermediately local powering-invariant in nilpotent group: Difference between revisions

Latest revision as of 01:43, 3 July 2026

Statement

Suppose $G$ is a nilpotent group and $H$ is a subgroup containing the center of $G$ that is also a local powering-invariant subgroup of $G$ . Then, $H$ is an intermediately local powering-invariant subgroup of $G$ . Explicitly, suppose $K$ is a subgroup of $G$ containing $H$ . Then, $H$ is a local powering-invariant subgroup of $K$ .

Related facts

Local powering-invariant subgroup containing the center is intermediately powering-invariant in nilpotent group

Facts used

Torsion-freeness for a prime is subgroup-closed
Equivalence of definitions of nilpotent group that is torsion-free for a set of primes
Nilpotency is subgroup-closed

Proof

Given: A nilpotent group $G$ , a subgroup $H$ of $G$ that is local powering-invariant and such that $Z (G) \leq H$ where $Z (G)$ is the center of $G$ . A subgroup $K$ of $G$ containing $H$ . A prime number $p$ and an element $h \in H$ such that there is a unique element $x \in K$ satisfying $x^{p} = h$ .

To prove: There exists a unique element $x \in H$ such that $x^{p} = h$ .

Proof:

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	$Z (G) \leq K$ .		$Z (G) \leq H, H \leq K$		given-direct
2	$K$ is nilpotent.	Fact (3)	$G$ is nilpotent, $K \leq G$		Given-fact direct
3	$K$ is $p$ -torsion-free.	Fact (2)	$h \in K$ has a unique $p^{t h}$ root in $K$		We use the equivalence (3) implies (1) within the multi-part equivalence of Fact (2).
4	$Z (G)$ is $p$ -torsion-free.	Fact (1)		Steps (1), (3)	Step-fact combination direct
5	The map $t \mapsto t^{p}$ is injective in $G$ .	Fact (2)	$G$ is nilpotent	Step (4)	Step-fact combination direct (specifically, we want to use the implication from (4) to (1) in the multi-part equivalence of Fact (2))
6	The element $x \in K$ is the unique $p^{t h}$ root of $h$ in all of $G$ .		$x \in K$ satisfies $x^{p} = h$	Step (5)	given-step direct
7	The element $x$ of Step (6) is in $H$ .		$H$ is local powering-invariant in $G$	Step (6)	Step-given combination direct.

@@ Line 1: / Line 1: @@
 ==Statement==
-Suppose <math>G</math> is a [[nilpotent group]] and <math>H</math> is a [[subgroup containing the center]] of <math>G</math> that is also a [[local powering-invariant subgroup]] of <math>G</math>. Then, <math>H</math> is an [[intermediately local powering-invariant subgroup]] of <math>G</math>. Explicitly, suppose <math>K</math> is a subgroup of <math>G</math> containing <math>H</math>. Then, <math>H</math> is a [[local powering-invariant subgroup]] of <math>K</math>.
+Suppose <math>G</math> is a [[fact about::nilpotent group;3| ]][[uses property satisfaction of::nilpotent group]] and <math>H</math> is a [[subgroup containing the center]] of <math>G</math> that is also a [[fact about::local powering-invariant subgroup;2| ]][[uses property satisfaction of::local powering-invariant subgroup]] of <math>G</math>. Then, <math>H</math> is an [[fact about::intermediately local powering-invariant subgroup;3| ]][[proves property satisfaction of::intermediately local powering-invariant subgroup]] of <math>G</math>. Explicitly, suppose <math>K</math> is a subgroup of <math>G</math> containing <math>H</math>. Then, <math>H</math> is a [[local powering-invariant subgroup]] of <math>K</math>.
 ==Related facts==
@@ Line 11: / Line 11: @@
 # Torsion-freeness for a prime is subgroup-closed
 # [[uses::Equivalence of definitions of nilpotent group that is torsion-free for a set of primes]]
+# [[uses::Nilpotency is subgroup-closed]]
 ==Proof==
@@ Line 23: / Line 24: @@
 ! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation
 |-
-| 1 || <math>Z(G) \le K</math>. || || <math>Z(G) \le H, H \le K</math>. || || given-direct
+| 1 || <math>Z(G) \le K</math>. || || <math>Z(G) \le H, H \le K</math> || || given-direct
 |-
-| 2 || <math>K</math> is <math>p</math>-torsion-free. || Fact (2) || <math>h \in K</math> has a unique <math>p^{th}</math> root in <math>K</math>. || || We use the equivalence (3) implies (1) within the multi-part equivalence of Fact (2).
+| 2 || <math>K</math> is nilpotent. || Fact (3) || <math>G</math> is nilpotent, <math>K \le G</math> || || Given-fact direct
 |-
-| 3 || <math>Z(G)</math> is <math>p</math>-torsion-free. || Fact (1) || || Steps (1), (2)  || Step-fact combination direct
+| 3 || <math>K</math> is <math>p</math>-torsion-free. || Fact (2) || <math>h \in K</math> has a unique <math>p^{th}</math> root in <math>K</math> || || We use the equivalence (3) implies (1) within the multi-part equivalence of Fact (2).
 |-
-| 4 || The map <math>t \mapsto t^p</math> is injective in <math>G</math>. || Fact (2) ||<math>G</math> is nilpotent || Step (3) || Step-fact combination direct (specifically, we want to use the implication from (4) to (1) in the multi-part equivalence of Fact (2))
+| 4 || <math>Z(G)</math> is <math>p</math>-torsion-free. || Fact (1) || || Steps (1), (3)  || Step-fact combination direct
 |-
-| 5 || The element <math>x \in K</math>is the unique <math>p^{th}</math> root of <math>h</math> in all of <math>G</math>. || || <math>x \in K</matH> satisfies <math>x^p = h</math>. || Step (4) || given-step direct
+| 5 || The map <math>t \mapsto t^p</math> is injective in <math>G</math>. || Fact (2) ||<math>G</math> is nilpotent || Step (4) || Step-fact combination direct (specifically, we want to use the implication from (4) to (1) in the multi-part equivalence of Fact (2))
 |-
-| 6 || The element <math>x</math> of Step (5) is in <math>H</math>. || || <math>H</math> is local powering-invariant in <math>G</math>. || Step (5) || Step-given combination direct.
+| 6 || The element <math>x \in K</math>is the unique <math>p^{th}</math> root of <math>h</math> in all of <math>G</math>. || || <math>x \in K</matH> satisfies <math>x^p = h</math> || Step (5) || given-step direct
+|-
+| 7 || The element <math>x</math> of Step (6) is in <math>H</math>. || || <math>H</math> is local powering-invariant in <math>G</math> || Step (6) || Step-given combination direct.
 |}