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A subgroup is essentially a smaller group within a larger group. For the quaternion group, denoted as \(Q_8\), we begin by identifying the elements of the group. \(Q_8\) consists of the elements \(\{1, -1, i, -i, j, -j, k, -k\}\). Each of these elements can generate a cyclic subgroup, which is a minimal subgroup containing that element.

When listing cyclic subgroups, it's essential to recognize that some elements, such as \(-i\), \(-j\), and \(-k\), generate subgroups that are identical to those generated by \(i\), \(j\), and \(k\) respectively. So, to avoid repetition, we only keep distinct subgroups. After this elimination, the significant subgroups are:

• \(\langle 1 \rangle = \{1\}\)
• \(\langle -1 \rangle = \{-1, 1\}\)
• \(\langle i \rangle = \{i, -i, 1, -1\}\)
• \(\langle j \rangle = \{j, -j, 1, -1\}\)
• \(\langle k \rangle = \{k, -k, 1, -1\}\)

These subgroups show how different subsets of \(Q_8\) can still retain the structure of a group.

A normal subgroup is a special type of subgroup that remains invariant under conjugation by any element of the group. In simpler terms, if \(N\) is a normal subgroup of a group \(G\), then every element of \(G\) can "shuffle" the elements of \(N\) without changing the subgroup itself. Mathematically, this is expressed as \(gNg^{-1} = N\) for every \(g \in G\).

In the quaternion group \(Q_8\), we determine which of its subgroups are normal. Here, \(\langle 1 \rangle = \{1\}\) is trivially normal because it only contains the identity element, which does not change under group operations. Similarly, \(\langle -1 \rangle = \{-1, 1\}\) is also normal since conjugation by any element of \(Q_8\) does not result in elements outside of \{-1, 1\}.

• \(\langle 1 \rangle = \{1\}\)
• \(\langle -1 \rangle = \{-1, 1\}\)

These are the only normal subgroups of \(Q_8\), influencing possible factor groups derived from \(Q_8\).

Factor groups, also known as quotient groups, emerge when a normal subgroup is used to partition the original group into distinct sets called cosets. Each coset serves as a single element in the new group. Factor groups essentially provide a way of simplifying the group's structure.

For \(Q_8\), the factor groups are determined through its normal subgroups. The first factor group \(Q_8 / \{1\}\) remains isomorphic to \(Q_8\) since the only subgroup is trivial. The second factor group, \(Q_8 / \{-1, 1\}\), results in cosets that significantly simplify the group structure, reflecting the Klein four-group structure. This is because:

• The cosets are \(\{1,-1\}, \{i,-i\}, \{j,-j\}, \{k,-k\}\).
• Each coset, just like each non-identity element in the Klein four-group, has an order of 2.

Consequently, the factor groups of \(Q_8\), up to isomorphism, are the quaternion group itself and the Klein four-group. This offers insight into the fundamental structure and properties of \(Q_8\).