Subgroups of order 4 in groups of order 8

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This article contains summary information about all subgroups of order 4 inside groups of order 8. See also groups of order 8 | groups of order 4 | subgroup structure of groups of order 8 | supergroups of groups of order 4

This article gives information on the occurrence of groups of order 4 as subgroups inside groups of order 8.

There are two groups of order 4: cyclic group:Z4 and Klein four-group. There are five groups of order 8: cyclic group:Z8, direct product of Z4 and Z2, dihedral group:D8, quaternion group, and elementary abelian group:E8.

Orderuptoeightbysubgroupinclusion.png

List of all subgroups

We make some preliminary observation: Any subgroup of order 4 in a group of order 8 must have index two. We have index two implies normal, so the subgroup is normal and the quotient group is isomorphic to cyclic group:Z2.

In the table below, we describe, for each isomorphism class of group of order 4 and group of order 8, all the possible automorphism classes of ways in which the group of order 4 occurs as a subgroup of the group of order 8. It turns out that in all cases, there is at most one automorphism class, i.e., all subgroups of order 4 in groups of order 8 are isomorph-automorphic subgroups (this feature is due to the small order, and does not carry over to higher orders).

Group GAP ID 2nd part Hall-Senior number Hall-Senior symbol Nilpotency class Occurrences of cyclic group:Z4 Number of subgroups in each automorphism class Occurrences of Klein four-group Number of subgroups in each automorphism class
cyclic group:Z8 1 3 (3) 1 Z4 in Z8 1 -- --
direct product of Z4 and Z2 2 2 (21) 1 Z4 in direct product of Z4 and Z2 2 first omega subgroup of direct product of Z4 and Z2 1
dihedral group:D8 3 4 8\Gamma_2a_1 2 cyclic maximal subgroup of dihedral group:D8 1 Klein four-subgroups of dihedral group:D8 2
quaternion group 4 5 8\Gamma_2a_2 2 cyclic maximal subgroups of quaternion group 3 -- --
elementary abelian group:E8 5 1 (1^3) 1 -- -- V4 in E8 7

Cyclic subgroups of order four

As noted above, there is at most one automorphism class for each isomorphism class of group of order 8 and subgroup of order 4. This feature is specific to the small orders involved.

Group GAP ID 2nd part Hall-Senior number Hall-Senior symbol Nilpotency class Occurrence of cyclic group:Z4 Number of subgroups = Number of normal subgroups = Number of conjugacy classes of subgroups (=1 iff characteristic subgroup) Size of orbit under automorphisms in the subgroup generated by 2-automorphisms (=1 iff the subgroup is a 2-automorphism-invariant subgroup) Size of orbit under automorphisms in the 2-core of the automorphism group (=1 iff the subgroup is a 2-core-automorphism-invariant subgroup) Size of orbit under automorphisms in the subgroup generated by 2'-automorphisms (=1 iff the subgroup is a coprime automorphism-invariant subgroup)
cyclic group:Z8 1 3 (3) 1 Z4 in Z8 1 1 1 1
direct product of Z4 and Z2 2 2 (21) 1 Z4 in direct product of Z4 and Z2 2 2 2 1
dihedral group:D8 3 4 8\Gamma_2a_1 2 cyclic maximal subgroup of dihedral group:D8 1 1 1 1
quaternion group 4 5 8\Gamma_2a_2 2 cyclic maximal subgroups of quaternion group 3 3 1 3
elementary abelian group:E8 5 1 (1^3) 1 -- -- -- -- --

Note the following significance for the last column: any subgroup of order 4 in a group of order 8 is already an isomorph-normal subgroup, and we know that isomorph-normal coprime automorphism-invariant implies Sylow-relatively weakly closed (more generally isomorph-normal coprime automorphism-invariant implies fusion system-relatively weakly closed). Thus, if the last column value is 1, then the subgroup is a weakly closed subgroup of the whole group relative to its occurrence as a Sylow subgroup in any bigger group.

Klein four-subgroups

Group GAP ID 2nd part Hall-Senior number Hall-Senior symbol Nilpotency class Occurrence of cyclic group:Z4 Number of subgroups = Number of normal subgroups = Number of conjugacy classes of subgroups (=1 iff characteristic subgroup) Size of orbit under automorphisms in the subgroup generated by 2-automorphisms (=1 iff the subgroup is a 2-automorphism-invariant subgroup) Size of orbit under automorphisms in the 2-core of the automorphism group (=1 iff the subgroup is a 2-core-automorphism-invariant subgroup)
cyclic group:Z8 1 3 (3) 1 -- -- -- --
direct product of Z4 and Z2 2 2 (21) 1 first omega subgroup of direct product of Z4 and Z2 1 1 1
dihedral group:D8 3 4 8\Gamma_2a_1 2 Klein four-subgroups of dihedral group:D8 2 2 2
quaternion group 4 5 8\Gamma_2a_2 2 -- -- -- --
elementary abelian group:E8 5 1 (1^3) 1 V4 in E8 7 7 1

Numerical information on counts of subgroups

The table below provides information on the counts of subgroups of order 4 in groups of order 8. Note the following:

General assertion Implication for the counts in this case
congruence condition on number of subgroups of given prime power order The number of subgroups of order 4 is odd.
The number of normal subgroups of order 4 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 4 is odd.
index two implies normal (or more generally, any maximal subgroup of a group of prime power order is normal and has prime index) The number of subgroups of order 4 equals the number of normal subgroups of order 4.
In a group of prime power order p^n, the maximal subgroups are precisely the subgroups of index p. They all contain the Frattini subgroup, and via the fourth isomorphism theorem, they correspond precisely to maximal subgroups of the Frattini quotient, which is elementary abelian of order p^r where r is the minimum size of generating set for the group. The number of maximal subgroups is (p^r - 1)/(p - 1). The number of subgroups of order 4 in a group of order 8 is 1,3, or 7, these values occurring when the minimum size of generating set is respectively 1, 2, and 3.
Group Second part of GAP ID Hall-Senior number Hall-Senior symbol Nilpotency class Minimum size of generating set Number of subgroups of order 4
(must be odd)
Number of normal subgroups of order 4
(must be odd)
Number of 2-core-automorphism-invariant subgroups of order 4
(must be odd)
Number of 2-automorphism-invariant subgroups of order 4 Number of characteristic subgroups of order 4
cyclic group:Z8 1 3 (3) 1 1 1 1 1 1 1
direct product of Z4 and Z2 2 2 (21) 1 2 3 3 1 1 1
dihedral group:D8 3 4 8\Gamma_2a_1 2 2 3 3 1 1 1
quaternion group 4 5 8\Gamma_2a_2 2 2 3 3 3 0 0
elementary abelian group:E8 5 1 (1^3) 1 3 7 7 7 0 0

Failure of congruence condition and replacement on elementary abelian groups

By the Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime, it is true that for an odd prime p, the singleton collection of the elementary abelian group of order p^2 is a collection of groups satisfying a universal congruence condition for the prime p, i.e., if there exists an elementary abelian subgroup of order p^2, the number of such subgroups is congruent to 1 mod p. In particular, the number of elementary abelian normal subgroups of order p^2 is congruent to 1 mod p. Further, the number of elementary abelian p-core-automorphism-invariant subgroups of order p^2 is also congruent to 1 mod p. Thus, in particular, the existence of an elementary abelian subgroup of order p^2 guarantees the existence of an elementary abelian normal subgroup of order p^2, and even the existence of an elementary abelian p-core-automorphism-invariant subgroup of order p^2.

However, this fails for the prime p = 2. Specifically, note that:

  • The congruence condition fails for dihedral group:D8, which has exactly two Klein four-subgroups, so the number of Klein four-subgroups is nonzero and not congruent to 1 mod 2.
  • Replacement of a Klein four-subgroup by a 2-core-automorphism-invariant Klein four-subgroup also fails for dihedral group:D8, which has a Klein four-subgroup but no 2-core-automorphism-invariant Klein four-subgroup.
  • Replacement by a normal subgroup holds in groups of order 8 for the silly reason that any subgroup of order 4 is already normal. However, we can use the failure of the stronger version of replacement to construct an example of order 16, namely dihedral group:D16, where replacement by a normal subgroup also fails.