Composition Series

Consider the group $G=\mathbb{Z}_{4}$.

1. Find a subnormal series of length 2 for $G$.

2. Hence find the factor groups of that subnormal series.

3. Explain whether the subnormal series found in (i) is a normal series.

Find the order of the maximal normal subgroup of $\mathbb{Z}_{120}$.

Find the two unique composition series for $\mathrm{C}_{6}$ (up to isomorphism), their composition lengths, and their composition factors (up to isomorphism).

Let $p$ be a prime. Show that any group with order $p^{2}$ has only one composition series up to isomorphism.

Consider the Klein four-group $\mathrm{V}$ with presentation

Recall that $\mathrm{V}\cong(\mathbb{Z}_{2})^{2}$.

1. Find the unique composition series for $\mathrm{V}$ up to isomorphism.

2. Let $\mathrm{Q}$ denote the quaternion group. Find the two distinct composition series for $\mathrm{Q}$ up to isomorphism.(Hint: consider the alternate definition of the quaternion group.)

3. Let $H$ be the subgroup of $\mathrm{Q}$ that is isomorphic to $\mathrm{C}_{4}$, and $K$ be a subgroup of $\mathrm{Q}$ that is isomorphic to $\mathrm{V}$.…

Find the unique composition series of $\mathrm{S}_{4}$, up to isomorphism.(Hint: consider the orders of elements in $\mathrm{A}_{4}$ and 1.8 to prove why a particular subgroup with a particular order cannot exist.)