Cyclic maximal subgroups of quaternion group
This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z4 and the group is (up to isomorphism) quaternion group (see subgroup structure of quaternion group).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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The quaternion group is defined by the following presentation:
We use the symbol for the identity element and the symbol for . are denoted by respectively.
It has the following multiplication table:
The subgroups that we are interested in studying are the three four-element subgroups of this group, namely the subgroups:
- 1 Invariance under automorphisms and endomorphisms: properties satisfied
- 2 Invariance under automorphisms and endomorphisms: properties not satisfied
- 3 Resemblance notions
- 4 Generic maximality notions
- 5 Maximality notions related to abelianness
- 6 GAP implementation
Invariance under automorphisms and endomorphisms: properties satisfied
All three of these subgroups are invariant under all retractions of the whole group. This is because of the trivial reason that the group has no retractions under that the identity map and the trivial homomorphism.
Invariance under automorphisms and endomorphisms: properties not satisfied
Further information: characteristic subgroup
None of these three subgroups is characteristic. In fact, there is an automorphism cyclically permuting that permutes these subgroups cyclically.
Coprime automorphism-invariant and cofactorial automorphism-invariant
None of these three subgroups is invariant either under the automorphisms whose order is a power of two or under the automorphisms whose order is relatively prime to two. Specifically:
- There is an automorphism of order three cyclically permuting .
- There is an automorphism of order two that sends to , to , and to . This has order two and it interchanges two of the three subgroups while sending the third subgroup to itself.
Generic maximality notions
All three subgroups are maximal subgroups of the group of prime power order. Thus, they satisfy all these properties: maximal normal subgroup, maximal subgroup, subgroup of index two, order-normal subgroup, isomorph-normal subgroup, maximal subgroup of finite nilpotent group.
Abelian subgroups of maximum order
Abelian subgroups of maximum rank
All three subgroups are abelian subgroups of maximum rank. However, they are not the only ones -- so is the center, which has order two.
Finding these subgroups in a black-box quaternion group
L := MaximalSubgroups(G);
Alternatively, we can use the NormalSubgroups function:
L := Filtered(NormalSubgroups(G),H -> Order(H) = 4);
The individual members are accessed as respectively.