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.
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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 $p$ and $p^2$) | 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) $\Gamma_2$.

For an abelian group of order 32 and rank $r$ (i.e., $r$ is the minimum size of generating set), the number of abelian normal subgroups of order 16 is $2^r - 1$, 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 $(5)$ 1 1 0 0 0 0 1 1
SmallGroup(32,2) 2 18 $\Gamma_2h$ 2 2 0 3 0 0 0 3
Direct product of Z8 and Z4 3 5 $(32)$ 1 2 0 0 1 2 0 3
Semidirect product of Z8 and Z4 of M-type 4 19 $\Gamma_2i$ 2 2 0 0 1 2 0 3
SmallGroup(32,5) 5 20 $\Gamma_2j_1$ 2 2 0 1 0 2 0 3
Faithful semidirect product of E8 and Z4 6 46 $\Gamma_7a_1$ 3 2 0 0 0 0 0 0
SmallGroup(32,7) 7 47 $\Gamma_7a_2$ 3 2 0 0 0 0 0 0
SmallGroup(32,8) 8 48 $\Gamma_7a_3$ 3 2 0 0 0 0 0 0
SmallGroup(32,9) 9 27 $\Gamma_3c_1$ 3 2 0 0 0 1 0 1
SmallGroup(32,10) 10 28 $\Gamma_3c_2$ 3 2 0 0 0 1 0 1
Wreath product of Z4 and Z2 11 31 $\Gamma_3e$ 3 2 0 0 1 0 0 1
SmallGroup(32,12) 12 21 $\Gamma_2j_2$ 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 $\Gamma_3f$ 3 2 0 0 0 1 0 1
Direct product of Z16 and Z2 16 6 $(41)$ 1 2 0 0 0 1 2 3
M32 17 22 $\Gamma_2k$ 2 2 0 0 0 1 2 3
Dihedral group:D32 18 49 $\Gamma_8a_1$ 4 2 0 0 0 0 1 1
Semidihedral group:SD32 19 50 $\Gamma_8a_2$ 4 2 0 0 0 0 1 1
Generalized quaternion group:Q32 20 51 $\Gamma_8a_3$ 4 2 0 0 0 0 1 1
Direct product of Z4 and Z4 and Z2 21 3 $(2^21)$ 1 3 0 3 4 0 0 7
Direct product of SmallGroup(16,3) and Z2 22 11 $\Gamma_2c_1$ 2 3 1 2 0 0 0 3
Direct product of SmallGroup(16,4) and Z2 23 12 $\Gamma_2c_2$ 2 3 0 3 0 0 0 3
SmallGroup(32,24) 24 16 $\Gamma_2f$ 2 3 0 1 2 0 0 3
Direct product of D8 and Z4 25 14 $\Gamma_2e_1$ 2 3 0 2 1 0 0 3
Direct product of Q8 and Z4 26 15 $\Gamma_2e_2$ 2 3 0 0 3 0 0 3
SmallGroup(32,27) 27 33 $\Gamma_4a_1$ 2 3 1 0 0 0 0 1
SmallGroup(32,28) 28 36 $\Gamma_4b_1$ 2 3 0 1 0 0 0 1
SmallGroup(32,29) 29 37 $\Gamma_4b_2$ 2 3 0 1 0 0 0 1
SmallGroup(32,30) 30 38 $\Gamma_4c_1$ 2 3 0 0 1 0 0 1
SmallGroup(32,31) 31 39 $\Gamma_4c_2$ 2 3 0 0 1 0 0 1
SmallGroup(32,32) 32 40 $\Gamma_4c_3$ 2 3 0 0 1 0 0 1
SmallGroup(32,33) 33 41 $\Gamma_4d$ 2 3 0 0 1 0 0 1
Generalized dihedral group for direct product of Z4 and Z4 34 34 $\Gamma_4a_2$ 2 3 0 0 1 0 0 1
SmallGroup(32,35) 35 35 $\Gamma_4a_3$ 2 3 0 0 1 0 0 1
Direct product of Z8 and V4 36 4 $(31^2)$ 1 3 0 1 0 6 0 7
Direct product of M16 and Z2 37 13 $\Gamma_2d$ 2 3 0 1 0 2 0 3
Central product of D8 and Z8 38 17 $\Gamma_2g$ 2 3 0 0 0 3 0 3
Direct product of D16 and Z2 39 23 $\Gamma_3a_1$ 3 3 0 0 0 1 0 1
Direct product of SD16 and Z2 40 24 $\Gamma_3a_2$ 3 3 0 0 0 1 0 1
Direct product of Q16 and Z2 41 25 $\Gamma_3a_3$ 3 3 0 0 0 1 0 1
Central product of D16 and Z4 42 26 $\Gamma_3b$ 3 3 0 0 0 1 0 1
Holomorph of Z8 43 44 $\Gamma_6a_1$ 3 3 0 0 0 0 0 0
SmallGroup(32,44) 44 45 $\Gamma_6a_2$ 3 3 0 0 0 0 0 0
Direct product of E8 and Z4 45 2 $(21^3)$ 1 4 1 14 0 0 0 15
Direct product of D8 and V4 46 8 $\Gamma_2a_1$ 2 4 2 1 0 0 0 3
Direct product of Q8 and V4 47 9 $\Gamma_2a_2$ 2 4 0 3 0 0 0 3
Direct product of SmallGroup(16,13) and Z2 48 10 $\Gamma_2b$ 2 4 0 3 0 0 0 3
Inner holomorph of D8 49 42 $\Gamma_5a_1$ 2 4 0 0 0 0 0 0
Central product of D8 and Q8 50 43 $\Gamma_5a_2$ 2 4 0 0 0 0 0 0
Elementary abelian group:E32 51 1 $(1^5)$ 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.

Group GAP ID (2nd part) Hall-Senior number Hall-Senior symbol Nilpotency class Minimum size of generating set Number of subgroups of order 16, exponent dividing 2 Number of subgroups of order 16, exponent dividing 4 Number of subgroups of order 16, exponent dividing 8 Number of subgroups of order 16, exponent dividing 16
Cyclic group:Z32 1 7 $(5)$ 1 1 0 0 0 1
SmallGroup(32,2) 2 18 $\Gamma_2h$ 2 2 0 3 3 3
Direct product of Z8 and Z4 3 5 $(32)$ 1 2 0 1 3 3
Semidirect product of Z8 and Z4 of M-type 4 19 $\Gamma_2i$ 2 2 0 1 3 3
SmallGroup(32,5) 5 20 $\Gamma_2j_1$ 2 2 0 1 3 3
Faithful semidirect product of E8 and Z4 6 46 $\Gamma_7a_1$ 3 2 0 0 0 0
SmallGroup(32,7) 7 47 $\Gamma_7a_2$ 3 2 0 0 0 0
SmallGroup(32,8) 8 48 $\Gamma_7a_3$ 3 2 0 0 0 0
SmallGroup(32,9) 9 27 $\Gamma_3c_1$ 3 2 0 0 1 1
SmallGroup(32,10) 10 28 $\Gamma_3c_2$ 3 2 0 0 1 1
Wreath product of Z4 and Z2 11 31 $\Gamma_3e$ 3 2 0 1 1 1
SmallGroup(32,12) 12 21 $\Gamma_2j_2$ 2 2 0 1 3 3
Semidirect product of Z8 and Z4 of semidihedral type 13 3 2 0 0 1 1
Semidirect product of Z8 and Z4 of dihedral type 14 3 2 0 0 1 1
SmallGroup(32,15) 15 32 $\Gamma_3f$ 3 2 0 0 1 1
Direct product of Z16 and Z2 16 6 $(41)$ 1 2 0 0 1 3
M32 17 22 $\Gamma_2k$ 2 2 0 0 1 3
Dihedral group:D32 18 49 $\Gamma_8a_1$ 4 2 0 0 0 1
Semidihedral group:SD32 19 50 $\Gamma_8a_2$ 4 2 0 0 0 1
Generalized quaternion group:Q32 20 51 $\Gamma_8a_3$ 4 2 0 0 0 1
Direct product of Z4 and Z4 and Z2 21 3 $(2^21)$ 1 3 0 7 7 7
Direct product of SmallGroup(16,3) and Z2 22 11 $\Gamma_2c_1$ 2 3 1 3 3 3
Direct product of SmallGroup(16,4) and Z2 23 12 $\Gamma_2c_2$ 2 3 0 3 3 3
SmallGroup(32,24) 24 16 $\Gamma_2f$ 2 3 0 1 3 3
Direct product of D8 and Z4 25 14 $\Gamma_2e_1$ 2 3 0 3 3 3
Direct product of Q8 and Z4 26 15 $\Gamma_2e_2$ 2 3 0 3 3 3
SmallGroup(32,27) 27 33 $\Gamma_4a_1$ 2 3 1 1 1 1
SmallGroup(32,28) 28 36 $\Gamma_4b_1$ 2 3 0 1 1 1
SmallGroup(32,29) 29 37 $\Gamma_4b_2$ 2 3 0 1 1 1
SmallGroup(32,30) 30 38 $\Gamma_4c_1$ 2 3 0 1 1 1
SmallGroup(32,31) 31 39 $\Gamma_4c_2$ 2 3 0 1 1 1
SmallGroup(32,32) 32 40 $\Gamma_4c_3$ 2 3 0 1 1 1
SmallGroup(32,33) 33 41 $\Gamma_4d$ 2 3 0 1 1 1
Generalized dihedral group for direct product of Z4 and Z4 34 34 $\Gamma_4a_2$ 2 3 0 1 1 1
SmallGroup(32,35) 35 35 $\Gamma_4a_3$ 2 3 0 1 1 1
Direct product of Z8 and V4 36 4 $(31^2)$ 1 3 0 1 7 7
Direct product of M16 and Z2 37 13 $\Gamma_2d$ 2 3 0 1 3 3
Central product of D8 and Z8 38 17 $\Gamma_2g$ 2 3 0 0 3 3
Direct product of D16 and Z2 39 23 $\Gamma_3a_1$ 3 3 0 0 1 1
Direct product of SD16 and Z2 40 24 $\Gamma_3a_2$ 3 3 0 0 1 1
Direct product of Q16 and Z2 41 25 $\Gamma_3a_3$ 3 3 0 0 1 1
Central product of D16 and Z4 42 26 $\Gamma_3b$ 3 3 0 0 1 1
Holomorph of Z8 43 44 $\Gamma_6a_1$ 3 3 0 0 0 0
SmallGroup(32,44) 44 45 $\Gamma_6a_2$ 3 3 0 0 0 0
Direct product of E8 and Z4 45 2 $(21^3)$ 1 4 1 15 15 15
Direct product of D8 and V4 46 8 $\Gamma_2a_1$ 2 4 2 3 3 3
Direct product of Q8 and V4 47 9 $\Gamma_2a_2$ 2 4 0 3 3 3
Direct product of SmallGroup(16,13) and Z2 48 10 $\Gamma_2b$ 2 4 0 3 3 3
Inner holomorph of D8 49 42 $\Gamma_5a_1$ 2 4 0 0 0 0
Central product of D8 and Q8 50 43 $\Gamma_5a_2$ 2 4 0 0 0 0
Elementary abelian group:E32 51 1 $(1^5)$ 1 5 31 31 31 31