Presentations for groups of order 8

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This article gives specific information, namely, presentations, about a family of groups, namely: groups of order 8.
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Power-commutator presentations

Each of the power-commutator presentations uses three generators (because \log_28 = 3) that we call a_1,a_2,a_3. The power relations are of the form a_1^2 = a_2^{\beta(1,2)} a_3^{\beta(1,3)} where \beta(1,2), \beta(1,3) are natural numbers that depend on the nature of the group, and a_2^2 = a_3^{\beta(2,3)} where \beta(2,3) depends on the nature of the group. The commutator relation is of the form [a_1,a_2] = a_3^{\beta(1,2,3)} where \beta(1,2,3) depends on the nature of the group. Note that a_3^2, [a_1,a_3], and [a_2,a_3] are always the identity.

The squaring comes because the underlying prime of the group is 2.

It turns out that for the isomorphism class of the final group, all the four values \beta(1,2),\beta(1,3),\beta(2,3),\beta(1,2,3) matter only mod the underlying prime, which in this case is 2. Thus, for simplicity, we assume that they are in the set \{ 0, 1 \}. Note that this is a general feature of power-commutator presentations.


Simplified power-commutator presentations

We here provide a single power-commutator presentation among the many possibilities.


Group Second part of GAP ID (GAP ID is (p^3,2nd part) Nilpotency class Minimum size of generating set Prime-base logarithm of exponent \beta(1,2) \beta(1,3) \beta(2,3) \beta(1,2,3) full power-commutator presentation
cyclic group:Z8 1 1 1 3 1 0 1 0 [SHOW MORE]
direct product of Z4 and Z2 2 1 2 2 0 1 0 0 [SHOW MORE]
dihedral group:D8 3 2 2 2 0 0 0 1 [SHOW MORE]
quaternion group 4 2 2 2 0 1 1 1 [SHOW MORE]
elementary abelian group:E8 5 1 3 1 0 0 0 0 [SHOW MORE]


Determining the isomorphism class from an arbitrary power-commutator presentation

Note that for the second and fourth groups, there are multiple sets of possible conditions, given in two separate rows within those groups. These are equivalent under permutations of the generators.

Note that because the prime is 2, and 0 and 1 are the only possibilities. Thus, saying "nonzero" means that the value must be 1.

Group Second part of GAP ID (GAP ID is (p^3,2nd part) Nilpotency class Minimum size of generating set Prime-base logarithm of exponent Condition on \beta(1,2) Condition on \beta(1,3) Condition on \beta(2,3) Condition on \beta(1,2,3) Total number of power-commutator presentations
cyclic group:Z8 1 1 1 3 nonzero arbitrary nonzero zero 2
direct product of Z4 and Z2 2 1 2 2 arbitrary
nonzero
zero
nonzero
nonzero
arbitrary
zero
zero
zero
zero
nonzero
nonzero
zero
zero
zero
zero
5
dihedral group:D8 3 2 2 2 arbitrary
arbitrary
zero
arbitrary
arbitrary
zero
nonzero 3
quaternion group 4 2 2 2 zero nonzero nonzero nonzero 1
elementary abelian group:E8 5 1 3 1 zero zero zero zero 1
not a group of order 8 (i.e., the presentation is not consistent) -- -- -- -- nonzero arbitrary arbitrary nonzero 4
Total -- -- -- -- -- -- -- -- 16 (equals 2^4)

Below, we list all the 16 possibilities for the 4-tuple and the corresponding group:

\beta(1,2) \beta(1,3) \beta(2,3) \beta(1,2,3) Group GAP ID 2nd part
0 0 0 0 elementary abelian group:E8 5
1 0 0 0 direct product of Z4 and Z2 2
0 1 0 0 direct product of Z4 and Z2 2
1 1 0 0 direct product of Z4 and Z2 2
0 0 1 0 direct product of Z4 and Z2 2
1 0 1 0 cyclic group:Z8 1
0 1 1 0 direct product of Z4 and Z2 2
1 1 1 0 cyclic group:Z8 1
0 0 0 1 dihedral group:D8 3
1 0 0 1 not a group of order 8 (actually becomes Klein four-group) --
0 1 0 1 dihedral group:D8 3
1 1 0 1 not a group of order 8 --
0 0 1 1 dihedral group:D8 3
1 0 1 1 not a group of order 8 --
0 1 1 1 quaternion group 4
1 1 1 1 not a group of order 8 --