arXiv:1802.03550v1 [] 10 Feb 2018

A note on a class of gyrogroups
Marius Tărnăuceanu
October 1, 2016
Abstract
In [1], a class of gyrogroups (G, ⊙) has been associated to certain
groups (G, ·). We give a necessary and sufficient condition for (G, ⊙) to
be gyrocommutative. We also prove that under a suitable assumption
two finite groups central by a 2-Engel group are isomorphic if and only
if their associated gyrogroups are isomorphic.

MSC (2010): Primary 20F99; Secondary 20D99, 20N05.
Key words: gyrogroup, gyrocommutative gyrogroup, gyrogroup isomorphism.

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Introduction

Gyrogroups are suitable generalization of groups, whose origin is described
in [7, 8]. They share remarkable analogies with groups. In fact, every group
forms a gyrogroup under the same operation. Many of classical theorems
in group theory also hold for gyrogroups, including the Lagrange theorem
[3], the fundamental isomorphism theorems [4], and the Cayley theorem [4]
(for all these theorems see also [6]). Gyrogroup actions and related results,
such as the orbit-stabilizer theorem, the orbit decomposition theorem, and
the Burnside lemma have been studied in [5].
The present note deals with a connection between groups and gyrogroups,
namely with the gyrogroup associated to any group central by a 2-Engel
group (see [1]). We determine conditions for such a gyrogroup to be gyrocommutative and for such two gyrogroups to be isomorphic. An interesting
conservative functor between a subcategory of groups and the category of
gyrogroups is constructed, as well.
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Recall that a groupoid (G, ⊙) is called a gyrogroup if its binary operation
satisfies the following axioms:
1. There is an element e ∈ G such that e ⊙ a = a for all a ∈ G.
2. For every a ∈ G, there is an element a′ ∈ G such that a′ ⊙ a = e.
3. For all a, b ∈ G, there is an automorphism gyr[a, b] ∈ Aut(G, ⊙) such
that
a ⊙ (b ⊙ c) = (a ⊙ b) ⊙ gyr[a, b](c)

(left gyroassociative law)

for all c ∈ G.
4. For all a, b ∈ G, gyr[a ⊙ b, b] = gyr[a, b].

(left loop property)

Moreover, if
a ⊙ b = gyr[a, b](b ⊙ a)

(gyrocommutative law)

for all a, b ∈ G, then (G, ⊙) is called a gyrocommutative gyrogroup.
We remark that the axioms in the above definition imply the right counterparts. In particular, any gyrogroup has a unique two-sided identity e, and
an element a of the gyrogroup has a unique two-sided inverse a′ . Given two
elements a, b of a gyrogroup G, the map gyr[a, b] is called the gyroautomorphism generated by a and b. By Theorem 2.10 of [7], the gyroautomorphisms
are completely determined by the gyrator identity
gyr[a, b](c) = (a ⊙ b)′ ⊙ [a ⊙ (b ⊙ c)]
for all a, b, c ∈ G. Obviously, every group forms a gyrogroup under the same
operation by defining the gyroautomorphisms to be the identity automorphism, but the converse is not in general true. From this point of view,
gyrogroups suitably generalize groups.
Recall also that gyrogroup homomorphism is a map between gyrogroups
that preserves the gyrogroup operations. A bijective gyrogroup homomorphism is called a gyrogroup isomorphism. We say that two gyrogroups G1
and G2 are isomorphic, written G1 ∼
= G2 , if there exists a gyrogroup isomorphism from G1 to G2 . Given a gyrogroup G, a gyrogroup isomorphism from
G to itself is called a gyrogroup automorphism of G.
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One of the most interesting purely algebraic classes of gyrogroups is introduced in [1], as follows. Define on a group (G, ·) the binary operation
a ⊙ b = a2 ba−1 , ∀ a, b ∈ G.
Then, by Theorem 3.7 of [1], we have:
Theorem 1. (G, ⊙) is a gyrogroup if and only if (G, ·) is central by a 2-Engel
group.
In what follows we will call (G, ⊙) the gyrogroup associated to a given
group (G, ·), which is assumed to be central by a 2-Engel group. Note that in
this case the gyroautomorphism generated by two elements a and b of G is
given by
gyr[a, b] = ϕ[a,b−1 ] ,
where ϕ[a,b−1 ] is the inner automorphism of G induced by the commutator
[a, b−1 ] of a and b−1 .
We are now in a position to characterize the gyrocommutativity of (G, ⊙).
Theorem 2. (G, ⊙) is gyrocommutative if and only if the inner automorphism group of (G, ·) is of exponent 3.
Clearly, if (G, ·) is commutative, then the binary operations · and ⊙ coincide, and (G, ⊙) is gyrocommutative. Note that there is also a non- commutative group, which is central by a 2-Engel group, such that its associated
gyrogroup is gyrocommutative (e.g. the group of upper triangular matrices
over F3 with diagonal (1,1,1)).
Next, let (G1 , ·) and (G2 , ·) be two finite groups central by a 2-Engel
group, and let (G1 , ⊙) and (G2 , ⊙) be their associated gyrogroups. Obviously, if (G1 , ·) ∼
= (G2 , ·), then a group isomorphism from G1 to G2 is also a
gyrogroup isomorphism from (G1 , ⊙) to (G2 , ⊙), that is, (G1 , ⊙) ∼
= (G2 , ⊙).
A sufficient condition for the converse to be true is given in the following
theorem.
Theorem 3. If 3 ∤ |G1 |, then (G1 , ·) ∼
=
= (G2 , ·) if and only if (G1 , ⊙) ∼
(G2 , ⊙).
From Theorem 3 we obtain the following corollary.

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Corollary 4. Let (G, ·) be a group central by a 2-Engel group such that
3 ∤ |G|. Then the group of all gyrogroup automorphisms of (G, ⊙) coincides
with the group of all automorphisms of (G, ·).
Finally, we observe that there is an interesting conservative functor F
between the category C of finite groups central by a 2-Engel group whose
order is not divisible by 3 and the category of gyrogroups, which associates
to each object (G, ·) in C the gyrogroup (G, ⊙) and to each homomorphism
f in C the gyrogroup homomorphism F (f ) = f .
Much of our notation is standard and will usually not be repeated here.
Elementary notions and results on groups can be found in [2].

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Proofs of the main results

Proof of Theorem 2. In our case the gyrocommutative law becomes
a2 ba−1 = ϕ[a,b−1 ] (b2 ab−1 ), ∀ a, b ∈ G,
which means
b3 a = ab3 , ∀ a, b ∈ G,
i.e.
b3 ∈ Z(G), ∀ b ∈ G.
Obviously, this condition is equivalent to exp(Inn(G)) = 3 in view of the
group isomorphism G/Z(G) ∼
= Inn(G).
Proof of Theorem 3. Assume that (G1 , ⊙) ∼
= (G2 , ⊙) and let f : G1 −→ G2
be a gyrogroup isomorphism. Then f is a bijection and
(1)

f (a2 ba−1 ) = f (a)2 f (b)f (a)−1 , ∀ a, b ∈ G.

If ei is the identity of Gi , i = 1, 2, we infer that
f (e1 ) = e2
by taking a = b = e1 in (1). Also, by taking b = a and b = a−1 in (1),
respectively, one obtains
f (a2 ) = f (a)2 and f (a−1 ) = f (a)−1 , ∀ a ∈ G1 .
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Next, let us write (1) with a−1 ba and a−1 b−1 instead of b, respectively. It
follows that
f (ab) = f (a)2 f (a−1 ba)f (a)−1

(2)
and
(3)

f (aba−1 ) = f (a)f (ba)f (a)−2 .

Replace a with a−1 in (3). Then
f (a−1 ba) = f (a)−1 f (ba−1 )f (a)2 ,
which together with (2) leads to
f (ab) = f (a)f (ba−1 )f (a).
By writing this equality with ba instead of b, we find
(4)

f (aba) = f (a)f (b)f (a), ∀ a, b ∈ G1 .

Finally, replacing b with a2 ba−1 in (4) gives
(5)

f (a3 b) = f (a)3 f (b), ∀ a, b ∈ G1 ,

and taking b = e1 in (5) gives
f (a3 ) = f (a)3 , ∀a ∈ G1 .

(6)

We are now in a position to prove that f is a group homomorphism. Let
x, y ∈ G1 . Since 3 ∤ |G1 |, we have 3 ∤ o(x) and consequently gcd(3, o(x)) = 1,
i.e. 1 = 3α + o(x)β for some integers α and β. It follows that
x = x3α+o(x)β = x3α xo(x)β = x3α .
Then (5) and (6) lead to
f (xy) = f (x3α y) = f ((xα )3 y) = f (xα )3 f (y) = f (x3α )f (y) = f (x)f (y),
as desired. Hence f is a group isomorphism, completing the proof.

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Marius Tărnăuceanu
Faculty of Mathematics
“Al.I. Cuza” University
Iaşi, Romania
e-mail: tarnauc@uaic.ro

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