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Worksheet 10.3.1 Activity: (Sub)Normal Series
A4US

Given a group, we can look at subgroups. We say that a sequence of subgroups

\begin{equation*} G = H_n \supset H_{n-1} \supset \cdots \supset H_1 \supset H_0 = \{e\} \end{equation*}

is a subnormal series provided each \(H_i\) is normal in \(H_{i+1}\text{,}\) and a normal series if each \(H_i\) is normal in \(G\text{.}\)

A non-trivial group \(G\) is called simple provided it has no non-trivial normal subgroups. We say that a subnormal series is a composition series and that a normal series is a principle series if every quotient group \(H_{i+1}/H_i\) is simple.

1.

Find a subnormal series for \(D_4\text{.}\) Is it a normal series?

2.

Find two different normal series for \(\Z_{60}\) of length 3 (length is the number of proper inclusions).

3.

Find the quotient groups \(H_{i+1}/H_i\) for both series above. How are these related? Are the series composition series?

4.

Find a composition series for \(\Z_{60}\text{.}\) Can you take it to be a refinement of the normal series you found above?