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| {{particular group}}
| | #redirect [[Modular maximal-cyclic group:M16]] |
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| ==Definition==
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| The group, sometimes denoted <math>M_{16}</math> or <math>M_4(2)</math>, is defined as follows:
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| <math>M_{16} = \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^5 \rangle</math>.
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| Here, <math>e</math> denotes the identity element.
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| ==Position in classifications==
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| {{quotation|Get more information about groups of the same order at [[Groups of order 16#The list]]}}
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| {| class="sortable" border="1"
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| ! Type of classification !! Position/number in classification
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| |-
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| | GAP ID || <math>(16,6)</math>, i.e., <math>6^{th}</math> among groups of order 16
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| |-
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| | Hall-Senior number || 12 among groups of order 16
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| |-
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| | Hall-Senior symbol || <math>16\Gamma_2d</math>
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| |}
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| ==Arithmetic functions==
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| {{M-type 2-group arithmetic function table|
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| order = 16|
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| order p-log = 4|
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| degree = 8|
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| degree p-log = 3}}
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| ==Group properties==
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| {| class="sortable" border="1"
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| !Property !! Satisfied? !! Explanation !! Comment
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| | {{group properties because p-group}}
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| |[[Dissatisfies property::Abelian group]] || No || <math>a,x</math> do not commute ||
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| |[[Satisfies property::Metacyclic group]] || Yes || ||
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| |[[Satisfies property::Finite group that is 1-isomorphic to an abelian group]] || Yes || via [[cocycle halving generalization of Baer correspondence]] || See [[element structure of groups of order 16#1-isomorphism]]
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| |}
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| ==Elements==
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| {{further|[[element structure of M16]]}}
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| ===Conjugacy class structure===
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| {{#lst:element structure of M16|conjugacy and automorphism class structure}}
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| ===1-isomorphism===
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| The group is [[1-isomorphic groups|1-isomorphic]] to the group [[direct product of Z8 and Z2]]. In other words, there is a bijection between the groups that restricts to an isomorphism on all cyclic subgroups on either side. The 1-isomorphism is explained by the [[cocycle halving generalization of Baer correspondence]], where the intermediary is a [[class two Lie cring]].
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| ==Subgroups==
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| {{further|[[subgroup structure of M16]]}}
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| To describe subgroups, we use the defining presentation given at the beginning:
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| <math>M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle</math>.
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| {{#lst:subgroup structure of M16|summary}}
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| ==Distinguishing features==
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| ===Smallest of its kind====
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| * This is a minimum order example of a non-abelian [[finite group that is 1-isomorphic to an abelian group]] -- it is 1-isomorphic to [[direct product of Z8 and Z2]]. It is, however, not the only such example: the other example is [[central product of D8 and Z4]]. See [[element structure of groups of order 16#1-isomorphism]] for more details.
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| * This is a minimum order example of a [[nilpotent group]] that is not a [[UL-equivalent group]], i.e., the [[upper central series]] and [[lower central series]] are not the same. However, it is not the only example. In fact, all groups of order 16 and class two share this property with it. The other examples are [[SmallGroup(16,3)]], [[nontrivial semidirect product of Z4 and Z4]], [[direct product of D8 and Z2]], [[direct product of Q8 and Z2]], and [[central product of D8 and Z4]].
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| * This is a minimum order example of a situation where a group has two [[characteristic subgroup]]s that are both isomorphic to each other but are distinct. Both these are cyclic subgroups of order four. It is, however, not the only example of this order; there are other examples where a similar behavior occurs, albeit for different orders of subgroups -- for instance, order 2 (in [[nontrivial semidirect product of Z4 and Z4]]).
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| ==GAP implementation==
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| {{GAP ID|16|6}}
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| ===Description by presentation===
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| The group can be defined using a presentation as follows:
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| <pre>gap> F := FreeGroup(2);;
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| gap> G := F/[F.1^8,F.2^2,F.2*F.1*F.2*F.1^(-5)];
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| <fp group on the generators [ f1, f2 ]>
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| gap> IdGroup(G);
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| [ 16, 6 ]</pre>
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