Abelian subgroup structure of groups of order 32
This article gives specific information, namely, abelian subgroup structure, about a family of groups, namely: groups of order 32.
View abelian subgroup structure of group families | View abelian subgroup structure of groups of a particular order |View other specific information about groups of order 32
Summary on existence, congruence, and replacement
Order of subgroup | Existence of abelian subgroup of that order guaranteed? | Existence of abelian normal subgroup of that order guaranteed? | Abelian-to-normal replacement for that order guaranteed? | Number of abelian subgroups of that order always odd if nonzero? |
---|---|---|---|---|
1 | Yes | Yes | Yes | Yes |
2 | Yes | Yes | Yes | Yes |
4 | Yes | Yes | Yes | Yes |
8 | Yes | Yes | Yes | Yes |
16 | No | No | Yes | Yes |
32 | No | No | Yes | Yes |
Abelian normal subgroups
FACTS TO CHECK AGAINST for abelian normal subgroup of group of prime power order:
EXISTENCE: existence of abelian normal subgroups of small prime power order
CONGRUENCE, IMPLIES ABELIAN-TO-NORMAL REPLACEMENT: congruence condition on number of subgroups of given prime power order (covers cases of and ) | congruence condition on number of abelian subgroups of prime-cube order | congruence condition on number of abelian subgroups of prime-fourth order | Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order | congruence condition on number of abelian subgroups of prime index | congruence condition on number of abelian subgroups of prime-square index for odd prime |abelian-to-normal replacement theorem for prime-square index | abelian-to-normal replacement theorem for prime-cube index for odd prime
CONGRUENCE CONDITIONS BASED ON EXPONENT: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
RELATION WITH REPRESENTATION THEORY: degree of irreducible representation divides index of abelian normal subgroup
Abelian normal subgroups of order 16
Note that index two implies normal, so the abelian subgroups of order 16 are precisely the same as the abelian normal subgroups of order 16.
- Existence is not guaranteed!: It is possible for there to be no abelian normal subgroup of order 16.
- Count: The congruence condition on number of abelian subgroups of prime index tells us that the number of abelian normal subgroups of order 16, if nonzero, is odd. However, we can be more specific. In a non-abelian group of order 32, the number of abelian normal subgroups is 0, 1, or 3. The "3" case arises if and only if the inner automorphism group is a Klein four-group, which occurs only for the Hall-Senior family (isoclinism class) .
For an abelian group of order 32 and rank (i.e., is the minimum size of generating set), the number of abelian normal subgroups of order 16 is , which could be 1, 3, 7, 15, or 31.
Group | GAP ID (2nd part) | Hall-Senior number | Hall-Senior symbol | Nilpotency class | Minimum size of generating set | Number of subgroups isomorphic to elementary abelian group:E16 | Number of subgroups isomorphic to direct product of Z4 and V4 | Number of subgroups isomorphic to direct product of Z4 and Z4 | Number of subgroups isomorphic to direct product of Z8 and Z2 | Number of subgroups isomorphic to cyclic group:Z16 | Total number of abelian (normal) subgroups of order 16 |
---|---|---|---|---|---|---|---|---|---|---|---|
Cyclic group:Z32 | 1 | 7 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | |
SmallGroup(32,2) | 2 | 18 | 2 | 2 | 0 | 3 | 0 | 0 | 0 | 3 | |
Direct product of Z8 and Z4 | 3 | 5 | 1 | 2 | 0 | 0 | 1 | 2 | 0 | 3 | |
Semidirect product of Z8 and Z4 of M-type | 4 | 19 | 2 | 2 | 0 | 0 | 1 | 2 | 0 | 3 | |
SmallGroup(32,5) | 5 | 20 | 2 | 2 | 0 | 1 | 0 | 2 | 0 | 3 | |
Faithful semidirect product of E8 and Z4 | 6 | 46 | 3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | |
SmallGroup(32,7) | 7 | 47 | 3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | |
SmallGroup(32,8) | 8 | 48 | 3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | |
SmallGroup(32,9) | 9 | 27 | 3 | 2 | 0 | 0 | 0 | 1 | 0 | 1 | |
SmallGroup(32,10) | 10 | 28 | 3 | 2 | 0 | 0 | 0 | 1 | 0 | 1 | |
Wreath product of Z4 and Z2 | 11 | 31 | 3 | 2 | 0 | 0 | 1 | 0 | 0 | 1 | |
SmallGroup(32,12) | 12 | 21 | 2 | 2 | 0 | 0 | 1 | 2 | 0 | 3 | |
Semidirect product of Z8 and Z4 of semidihedral type | 13 | 3 | 2 | 0 | 0 | 0 | 1 | 0 | 1 | ||
Semidirect product of Z8 and Z4 of dihedral type | 14 | 3 | 2 | 0 | 0 | 0 | 1 | 0 | 1 | ||
SmallGroup(32,15) | 15 | 32 | 3 | 2 | 0 | 0 | 0 | 1 | 0 | 1 | |
Direct product of Z16 and Z2 | 16 | 6 | 1 | 2 | 0 | 0 | 0 | 1 | 2 | 3 | |
M32 | 17 | 22 | 2 | 2 | 0 | 0 | 0 | 1 | 2 | 3 | |
Dihedral group:D32 | 18 | 49 | 4 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | |
Semidihedral group:SD32 | 19 | 50 | 4 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | |
Generalized quaternion group:Q32 | 20 | 51 | 4 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | |
Direct product of Z4 and Z4 and Z2 | 21 | 3 | 1 | 3 | 0 | 3 | 4 | 0 | 0 | 7 | |
Direct product of SmallGroup(16,3) and Z2 | 22 | 11 | 2 | 3 | 1 | 2 | 0 | 0 | 0 | 3 | |
Direct product of SmallGroup(16,4) and Z2 | 23 | 12 | 2 | 3 | 0 | 3 | 0 | 0 | 0 | 3 | |
SmallGroup(32,24) | 24 | 16 | 2 | 3 | 0 | 1 | 2 | 0 | 0 | 3 | |
Direct product of D8 and Z4 | 25 | 14 | 2 | 3 | 0 | 2 | 1 | 0 | 0 | 3 | |
Direct product of Q8 and Z4 | 26 | 15 | 2 | 3 | 0 | 0 | 3 | 0 | 0 | 3 | |
SmallGroup(32,27) | 27 | 33 | 2 | 3 | 1 | 0 | 0 | 0 | 0 | 1 | |
SmallGroup(32,28) | 28 | 36 | 2 | 3 | 0 | 1 | 0 | 0 | 0 | 1 | |
SmallGroup(32,29) | 29 | 37 | 2 | 3 | 0 | 1 | 0 | 0 | 0 | 1 | |
SmallGroup(32,30) | 30 | 38 | 2 | 3 | 0 | 0 | 1 | 0 | 0 | 1 | |
SmallGroup(32,31) | 31 | 39 | 2 | 3 | 0 | 0 | 1 | 0 | 0 | 1 | |
SmallGroup(32,32) | 32 | 40 | 2 | 3 | 0 | 0 | 1 | 0 | 0 | 1 | |
SmallGroup(32,33) | 33 | 41 | 2 | 3 | 0 | 0 | 1 | 0 | 0 | 1 | |
Generalized dihedral group for direct product of Z4 and Z4 | 34 | 34 | 2 | 3 | 0 | 0 | 1 | 0 | 0 | 1 | |
SmallGroup(32,35) | 35 | 35 | 2 | 3 | 0 | 0 | 1 | 0 | 0 | 1 | |
Direct product of Z8 and V4 | 36 | 4 | 1 | 3 | 0 | 1 | 0 | 6 | 0 | 7 | |
Direct product of M16 and Z2 | 37 | 13 | 2 | 3 | 0 | 1 | 0 | 2 | 0 | 3 | |
Central product of D8 and Z8 | 38 | 17 | 2 | 3 | 0 | 0 | 0 | 3 | 0 | 3 | |
Direct product of D16 and Z2 | 39 | 23 | 3 | 3 | 0 | 0 | 0 | 1 | 0 | 1 | |
Direct product of SD16 and Z2 | 40 | 24 | 3 | 3 | 0 | 0 | 0 | 1 | 0 | 1 | |
Direct product of Q16 and Z2 | 41 | 25 | 3 | 3 | 0 | 0 | 0 | 1 | 0 | 1 | |
Central product of D16 and Z4 | 42 | 26 | 3 | 3 | 0 | 0 | 0 | 1 | 0 | 1 | |
Holomorph of Z8 | 43 | 44 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | |
SmallGroup(32,44) | 44 | 45 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | |
Direct product of E8 and Z4 | 45 | 2 | 1 | 4 | 1 | 14 | 0 | 0 | 0 | 15 | |
Direct product of D8 and V4 | 46 | 8 | 2 | 4 | 2 | 1 | 0 | 0 | 0 | 3 | |
Direct product of Q8 and V4 | 47 | 9 | 2 | 4 | 0 | 3 | 0 | 0 | 0 | 3 | |
Direct product of SmallGroup(16,13) and Z2 | 48 | 10 | 2 | 4 | 0 | 3 | 0 | 0 | 0 | 3 | |
Inner holomorph of D8 | 49 | 42 | 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | |
Central product of D8 and Q8 | 50 | 43 | 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | |
Elementary abelian group:E32 | 51 | 1 | 1 | 5 | 31 | 0 | 0 | 0 | 0 | 31 |
We now construct a table derived from the above, that lists the total number of abelian normal subgroups of order eight and exponent bounded by some specific number (2, 4, 8, or 16). Note that the exponent dividing 2 count may be a nonzero even number (specifically, it can be 2), but the exponent dividing 4, exponent dividing 8, and exponent dividing 16 counts are all either zero or odd. As noted earlier, for non-abelian groups, these counts are either 0, 1, or 3.
Abelian normal subgroups of order 8
- Existence: There exist abelian normal subgroups of order 8 by the existence of abelian normal subgroups of small prime power order. In fact, any subgroup that is maximal among abelian normal subgroups must have orer at least 8.
- Count: The congruence condition on number of abelian subgroups of prime-cube order tells us that the number of abelian normal subgroups of order 8 is odd.
Abelian normal subgroups of order 4
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Abelian normal subgroups of order 2
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Abelian characteristic subgroups
Template:Abelian characteristic subgroups of finite p-group facts to check against
We make cases based on the nilpotency class, isoclinism class (i.e., Hall-Senior family) and other aspects of the group structure:
- For an abelian group of order 32, the whole group is the unique subgroup of itself that is maximal among abelian characteristic subgroups.