Holomorph of Z8: Difference between revisions

Revision as of 14:38, 2 September 2009

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

This particular group is a finite group of order: 32

Definition

This group (which we shall call $G$ throughout) can be defined in either of these ways:

It is the holomorph of the cyclic group on eight elements. In other words, it is the direct product of the cyclic group on eight elements, with its automorphism group.
It is the holomorph of the ring $Z / 8 Z$ . In other words, it is the general affine group $G A (1, Z / 8 Z)$ .

GAP implementation

Group ID

This group has ID $43$ among all the groups of order $2^{5} = 32$ . Thus, it can be defined using GAP's SmallGroup function:

SmallGroup(32,43)

Other definitions

The group can be defined using GAP's AutomorphismGroup and SemidirectProduct functions. Here is a full code snippet:

gap> C := CyclicGroup(8);
<pc group of size 8 with 3 generators>
gap> SemidirectProduct(AutomorphismGroup(C),C);
<pc group with 5 generators>

This can be compressed by coding a function Holomorph for computing the holomorph of a group. With this function coded, we can use:

Holomorph(CyclicGroup(8))

@@ Line 8: / Line 8: @@
 * It is the [[holomorph]] of the cyclic group on eight elements. In other words, it is the direct product of the cyclic group on eight elements, with its automorphism group.
-* It is the [[holomorph of a ring|holomorph]] of the ring <math>\Z/8\Z</math>.
+* It is the [[holomorph of a ring|holomorph]] of the ring <math>\Z/8\Z</math>. In other words, it is the [[member of family::general affine group]] <math>GA(1,\Z/8\Z)</math>.
-==Group properties==
+==GAP implementation==
-{{solvable}}
+===Group ID===
-The group is solvable. In fact, it is [[metabelian group|metabelian]], because the additive group is an Abelian [[normal subgroup]] (isomorphic to <math>\Z/8\Z</math>) and the quotient is Abelian, isomorphic to the [[Klein four-group]].
+This group has ID <math>43</math> among all the groups of order <math>2^5 = 32</math>. Thus, it can be defined using GAP's [[GAP:SmallGroup|SmallGroup]] function:
-The commutator subgroup of <math>G</math> is not the whole of the additive group, though. It is only the even integers in the additive group.
+<pre>SmallGroup(32,43)</pre>
-==Endomorphisms==
+===Other definitions===
-==Subgroups==
+The group can be defined using GAP's [[GAP:AutomorphismGroup|AutomorphismGroup]] and [[GAP:SemidirectProduct|SemidirectProduct]] functions. Here is a full code snippet:
+<pre>gap> C := CyclicGroup(8);
+<pc group of size 8 with 3 generators>
+gap> SemidirectProduct(AutomorphismGroup(C),C);
+<pc group with 5 generators></pre>
+This can be compressed by coding a function [[GAP:Holomorph|Holomorph]] for computing the holomorph of a group. With this function coded, we can use:
+<pre>Holomorph(CyclicGroup(8))</pre>