Subgroups of order 8 in groups of order 16
From Groupprops
This article contains summary information about all subgroups of order 8 inside groups of order 16. See also groups of order 16 | groups of order 8 | subgroup structure of groups of order 16 | supergroups of groups of order 8
This article describes the occurence of groups of order 8 as subgroups inside groups of order 16.
Contents
Numerical information on counts of subgroups
Counts of all subgroups
General assertion | Implication for the counts in this case |
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congruence condition on number of subgroups of given prime power order | The number of subgroups of order 8 is odd. The number of normal subgroups of order 8 is odd. The number of p-core-automorphism-invariant subgroups (which in this case means the number of subgroups invariant under automorphisms in the 2-core of the automorphism group) of order 8 is odd. |
index two implies normal, or more generally, see equivalence of definitions of maximal subgroup of group of prime power order | The number of subgroups of order 8 equals the number of normal subgroups of order 8. |
formula for number of maximal subgroups of group of prime power order: In a finite -group with minimum size of generating set , the number of maximal subgroups is . | The number of subgroups of order 8 in a group of order 16 is 1, 3, 7, or 15, these values occurring when the minimum size of generating set is respectively 1, 2, 3, and 4. For a non-abelian group of order 16, the only possibilities are 3 and 7, corresponding to minimum size of generating set being 2 or 3. |
Counts of abelian subgroups
General assertion | Implication for the counts in this case |
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existence of abelian normal subgroups of small prime power order: A group of order contains at least one abelian normal subgroup of order if . | Any group of order 16 contains an abelian normal subgroup of order 8. |
index two implies normal | number of abelian subgroups of order 8 = number of abelian normal subgroups of order 8. |
congruence condition on number of abelian subgroups of prime-cube order | The number of abelian subgroups of order 8 is odd. The number of abelian normal subgroups of order 8 is odd. The number of abelian 2-core-automorphism-invariant subgroups of order 8 is odd. |
congruence condition on number of abelian subgroups of prime index. Even more spceifically, the following: in a non-abelian group of order , the number of abelian subgroups of index is 0, 1, or . In an abelian group, it is of the form where is the minimum size of generating set. | In a non-abelian group of order 16, the number of abelian subgroups of order 8 is 1 or 3. In an abelian group of rank , the number is . |