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ABSTRACT: This study examines the link between
female reproductive investment and infant survival in pigs and its
implications for later reproductive success. Though maternal investment
is known to influence offspring survival, the conditions under which it
affects infant survival have remained largely unexplored. We tested for
a two-way interaction between mating and infant deaths in a sample of
110 litters from the Friesian Purebred Pig Population in North Zealand,
Denmark. Our results suggest a negative effect of reproductive success
on infant survival and a difference in survival over time. Mothers
that had a greater number of mating replacements gave birth to piglets
that survived longer. The effect of offspring survival on mating
replacement numbers was only significant in females where mating was
hierarchically controlled. Therefore we conclude that if a female
reproduces, the health of her offspring influences her investment in
further reproduction. We discuss the implications of our results for
further studies and breeding strategies to improve reproductive
success.Q:
Nilpotent commutator and nilpotent filtration
I have the following lemma, but don’t quite understand how to proof it. Can you give me any help please, thanks:
Let $G$ be a group, $g\in G$, $\{1,g\}\subset N(G)$ and $N\subset N(G)$ is a nilpotent normal subgroup.
(1) show that $g^{ -1}\in N$.
(2) Let $N\subset N\lhd G$ be a (nilpotent) filter on $G$ with $G=\bigcup_{g\in G}gN$, show that $G=\bigcup_{g\in G}gNg^{ -1}$.
How to proof the proposition (2) that is there are some hints.
A:
Let $n_1,\ldots,n_k\in N$ be the unique nilpotent normal series of $N$ with $N=\langle n_1\rangle\langle n_2,n_1\rangle\langle n_3,\ldots,n_{k-1},n_k\rangle$. Then
$$\begin{align*}
(xg)n_1\ld
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