Centrally indecomposable group: Difference between revisions

Latest revision as of 01:49, 13 September 2009

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Definition

Symbol-free definition

A nontrivial group is said to be centrally indecomposable if it satisfies the following equivalent conditions:

It cannot be expressed as the central product of two proper subgroups.
Every proper nontrivial central factor is a central subgroup, i.e., it is contained in the center of the group.

Definition with symbols

A group $G$ is said to be a centrally indecomposable group if it satisfies the following equivalent conditions:

We cannot write $G = H * K$ , viz., as a central product for proper subgroups $H$ and $K$ of $G$ .
Every proper nontrivial central factor of $G$ is a central subgroup of $G$ , i.e., it is contained in the center $Z (G)$ .

Relation with other properties

Stronger properties

Simple group

Weaker properties

@@ Line 7: / Line 7: @@
 ===Symbol-free definition===
-A nontrivial [[group]] is said to be '''centrally indecomposable''' if it cannot be expressed as the central product of two proper subgroups.
+A nontrivial [[group]] is said to be '''centrally indecomposable''' if it satisfies the following equivalent conditions:
-Note that, for a [[centerless group]], this is equivalent to saying that there is no proper nontrivial [[central factor]]. However, for an group with a nontrivial center, the center itself is a central factor.
+# It cannot be expressed as the [[defining ingredient::central product]] of two proper subgroups.
+# Every proper nontrivial [[defining ingredient::central factor]] is a [[defining ingredient::central subgroup]], i.e., it is contained in the [[center]] of the group.
 ===Definition with symbols===
-A [[group]] <math>G</math> is said to be a centrally indecomposable group if we cannot write:
+A [[group]] <math>G</math> is said to be a centrally indecomposable group if it satisfies the following equivalent conditions:
-<math>G = H * K</math>
+# We cannot write <math>G = H * K</math>, viz., as a [[central product]] for proper subgroups <math>H</math> and <math>K</math> of <math>G</math>.
+# Every proper nontrivial [[central factor]] of <math>G</math> is a [[central subgroup]] of <math>G</math>, i.e., it is contained in the [[center]] <math>Z(G)</math>.
-viz., as a [[central product]] for proper subgroups <math>H</math> and <math>K</math> of <math>G</math>.
 ==Relation with other properties==