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I'm trying to understand the definition of polycyclic groups.

A solvable group $G$ has two equivalent definitions:

  1. $G$ has a subnormal series like $$G = H_n \rhd H_{n-1} \rhd \cdots \rhd H_0 = 1$$ s.t. each $H_{i-1}$ is normal in $H_i$ and $H_{i}/H_{i-1}$ is an abelian group for all $i \in \{1, \ldots, n\}$.

  2. $G$ has a normal series like $$G = H_n \rhd H_{n-1} \rhd \cdots \rhd H_0 = 1$$ s.t. each $H_i$ is normal in $G$ and $H_{i}/H_{i-1}$ is an abelian group for all $i \in \{1, \ldots, n\}$.

Now Wikipedia says a polycyclic group is a solvable group in which the factors $H_{i}/G_{i-1}$ are cyclic but there is no requirement that each $H_i$ be normal in $G$:

In another direction, a polycyclic group must have a normal series with each quotient cyclic, but there is no requirement that each $H_{i}$ be normal in $G$. As every finite solvable group is polycyclic, this can be seen as one of the key differences between the definitions.

I don't understand this. If each $H_i$ is not normal in $G$ then the group $G$ doesn't even satisfy the definition of solvable groups. Furthermore, in a normal series, each $H_i$ is normal in $G$ by definition (cf. this)!

Could someone please explain what I'm missing here?

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Confusion arises because the same term normal series is used differently in the literature. To also quote Wikipedia

If in addition each $A_i$ is normal in $G$, then the series is called a normal series, when this term is not used for the weaker sense, or an invariant series.

So there are at least two schools:

  1. Call the weaker case subnormal and the stronger case normal series
  2. Call the weaker case normal and the stronger case invariant series.

As a side-effect of the multi-authorship, it is perhaps not completely enforceable that Wikipedia agree upon one of the two schemes consistently (perhaps it would be best to use only subnormal and invariant and get rid of the ambiguous normal)

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  • $\begingroup$ Thanks! Note that I've edited the Wikipedia article now. $\endgroup$ – S.D. Apr 30 '20 at 7:50
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The link you provided is for supersolvable groups. If you look over at solvable groups, they handle the issue of equivalence of some definitions. Here's a link:"Solvable group - Wikipedia" https://en.m.wikipedia.org/wiki/Solvable_group

If this is not the issue your having, I think the answer lies in the fact that they explicitly say that polycyclic groups are solvable. So I don't really see what the problem is. Afterwards they tack on some additional things, but unless you can show there's a contradiction, it's fine.

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