definition of group as direct sum of finite number of normal subgroups
Topics
About: group
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of normal subgroup of group.
Target Context
- The reader will have a definition of group as direct sum of finite number of normal subgroups.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*G\): \(\in \{\text{ the groups }\}\)
\( \{G_1, ..., G_n\}\): \(\subseteq \{\text{ the normal subgroups of } G\}\)
//
Conditions:
\(\forall G_j \in \{G_1, ..., G_n\} (G_j \cap (G_1 ... G_{j - 1} \hat{G_j} G_{j + 1} ... G_n) = \{1\})\)
\(\land\)
\(G = G_1 ... G_n\)
//
2: Natural Language Description
Any group, \(G\), such that there are some normal subgroups, \(G_1, ..., G_n\), of \(G\) such that for each \(G_j \in \{G_1, ..., G_n\}\), \(G_j \cap (G_1 ... G_{j - 1} \hat{G_j} G_{j + 1} ... G_n) = \{1\}\) and \(G = G_1 ... G_n\)
3: Note
\(G\) does not need to be Abelian.
The name is "group as direct sum of finite number of normal subgroups" instead of just "direct sum of finite number of normal subgroups" because the direct sum does not create any new group but it is about judging an existing group as the direct sum.
Do not confuse this definition with 'direct sum of modules', which is about creating a new module from the constituent modules. As any Abelian group is a module, there can be the direct sum of some Abelian groups by the concept of 'direct sum of modules', but that direct sum of modules is not exactly the direct sum of the constituent groups by this definition: any element of \(G_1 \oplus G_2\) is of the form, \((p_1, p_2)\), while any element of \(G_1\), \(p_1\), is not of the form, so, is not any element of \(G_1 \oplus G_2\), so, \(G_1\) is not any subgroup of \(G_1 \oplus G_2\); in fact, \(G_1 \oplus G_2\) is the direct sum of \(G_1 \times \{1\}\) and \(\{1\} \times G_2\) by this definition.
Any Abelian group as the direct sum of any finite number of normal subgroups by this definition is not equal to but is 'groups - homomorphisms' isomorphic to the direct sum of the subgroups by 'direct sum of modules', by the proposition that any group as the direct sum of any finite number of normal subgroups is 'groups - homomorphisms' isomorphic to the direct product of the subgroups.
Whether this definition requires that \(G = G_1 ... G_n\) or requires that \(G\) is generated by \(\{G_1, ..., G_n\}\) does not make any difference, because \(G_1 ... G_n = G_{\sigma_1} ... G_{\sigma_n}\) for each permutation of \((1, ..., n)\), \(\sigma\), by the proposition that for any group, the product of any finite number of normal subgroups is commutative and is a normal subgroup, so, \(G_1 ... G_n\) is nothing but the group generated by \(\{G_1, ..., G_n\}\).
The order of \(\{G_1, ..., G_n\}\) does not make any difference, by the proposition that any group as direct sum of finite number of normal subgroups is the group as direct sum of any reordered and combined normal subgroups.
