# Powering-invariant over quotient-powering-invariant implies powering-invariant

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## Statement

Suppose $G$ is a group and $H,K$ are subgroups of $G$ such that $H \le K \le G$, $H$ is normal in $G$, and the following conditions hold:

• $H$ is a quotient-powering-invariant subgroup of $G$.
• $K/H$ is a powering-invariant subgroup of $G/H$.

Then, $K$ is a powering-invariant subgroup of $G$.

## Proof

Given: $G$ is a group and $H,K$ are subgroups of $G$ such that $H \le K \le G$, $H$ is normal in $G$, and the following conditions hold:

• $H$ is a quotient-powering-invariant subgroup of $G$.
• $K/H$ is a powering-invariant subgroup of $G/H$. $p$ is a prime number such that $G$ is powered over $p$. An element $g \in K$.

To prove: There exists $x \in K$ such that $x^p = g$.

Proof: Let $\varphi:G \to G/H$ be the quotient map.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 There exists $x \in G$ such that $x^p = g$. $G$ is $p$-powered. Given-direct.
2 $G/H$ is $p$-powered. $H$ is quotient-powering-invariant in $G$, $G$ is $p$-powered. Given-direct.
3 $(\varphi(x))^p = \varphi(g)$ and $\varphi(x)$ is the unique $p^{th}$ root of $\varphi(g)$ in $G/H$. Steps (1), (2) Applying the homomorphism $\varphi$ to Step (1) gives that $(\varphi(x))^p = \varphi(g)$. Since, by Step (2), $G/H$ is $p$-powered, $\varphi(x)$ must be the unique $p^{th}$ root.
4 $\varphi(x) \in K/H$. $K/H$ is powering-invariant in $G/H$. Steps (2), (3) By Step (2), $G/H$ is $p$-powered, hence $K/H$ is $p$-powered from the given information. Thus, the unique $p^{th}$ root of $\varphi(g)$ in $G/H$, which equals $\varphi(x)$,(from Step (3)) must be in $K/H$.
5 $x \in K$. $H \le K$ Step (4) Since $\varphi(x) \in K/H$, $x \in \varphi^{-1}(K/H)$, which is $K$ since $H \le K$.