generalized quaternion group

generalized quaternion group

The groups given by the

presentation

Q

4

⁢

n

=

⟨

a

,

b

:

a

n

=

b

2

,

a

2

⁢

n

=

1

,

b

-

1

a

b

=

a

-

1

⟩

are the

generalized quaternion groups

. Generally one insists that

n

>

1

as the properties of generalized quaternions become more uniform at this stage.
However if

n

=

1

then one observes

a

=

b

2

so

Q

4

⁢

n

≅

ℤ

4

.

Dihedral group properties

are strongly related to generalized quaternion
group properties because of their highly related presentations. We will see this in many of our results.

Proposition 1

.

1.

|

Q

4

⁢

n

|

=

4

⁢

n

.

2.

Q

4

⁢

n

is

abelian

if and only if

n

=

1

.

3.

Every

element

x

∈

Q

4

⁢

n

can be written uniquely as

x

=

a

i

⁢

b

j

where

0

≤

i

<

2

⁢

n

and

j

=

0

,

1

.

4.

Z

⁢

(

Q

4

⁢

n

)

=

⟨

a

n

⟩

≅

ℤ

2

.

5.

Q

4

⁢

n

/

Z

⁢

(

Q

4

⁢

n

)

≅

D

2

⁢

n

.

Proof.

Given the

relation

b

-

1

⁢

a

⁢

b

=

a

-

1

(rather treating it as

a

⁢

b

=

b

⁢

a

-

1

) then
as with

dihedral groups

we can

shuffle

words in

{

a

,

b

}

to group all the

a

′

⁢

s

at the beginning and the

b

′

⁢

s

at the end. So every word takes the form

a

i

⁢

b

j

. As

|

a

|

=

2

⁢

n

and

|

b

|

=

4

we have

0

≤

i

<

2

⁢

n

and

0

≤

j

<

4

. However we have an
added relation that

a

n

=

b

2

so we can write

a

i

⁢

b

2

=

a

i

+

2

and
also

a

i

⁢

b

3

=

a

i

+

2

⁢

b

so we restrict to

j

=

0

,

1

. This gives us

4

⁢

n

elements
of this form which makes the order of

Q

4

⁢

n

at most

4

⁢

n

.

As

a

n

=

b

2

it follows

[

a

n

,

a

i

⁢

b

j

]

=

[

a

n

,

b

j

]

=

[

b

2

,

b

j

]

=

1

. So

a

n

is

central

. If we

quotient

by

⟨

a

n

⟩

then we have
the presentation

⟨

a

,

b

:

a

n

=

1

,

b

2

=

1

,

b

-

1

⁢

a

⁢

b

=

a

-

1

⟩

which we recognize as the presentation of the dihedral group. Thus

Q

4

⁢

n

/

⟨

a

n

⟩

≅

D

2

⁢

n

. This prove the order of

Q

4

⁢

n

is exactly

4

⁢

n

. Moreover, given

a

i

⁢

b

j

∈

Z

⁢

(

Q

4

⁢

n

)

we have

1

=

[

a

i

⁢

b

j

,

b

]

=

b

-

j

⁢

a

-

i

⁢

b

-

1

⁢

a

i

⁢

b

j

⁢

b

=

b

-

j

⁢

a

-

i

⁢

a

-

i

⁢

b

-

1

⁢

b

j

⁢

b

=

b

-

j

⁢

a

-

2

⁢

i

⁢

b

j

=

a

2

⁢

i

.

So we have

i

=

n

. So

a

n

⁢

b

j

=

b

j

+

2

. Then

1

=

[

b

j

+

2

,

a

]

forces

j

=

0

,

2

.
This means

Z

⁢

(

Q

4

⁢

n

)

=

⟨

a

n

⟩

=

⟨

b

2

⟩

.
∎

1

Examples

As mentioned, if

n

=

1

then

Q

4

≅

ℤ

4

. If

n

=

2

then we have the
usual

quaternion group

Q

8

. Because of the genesis of

quaternions

, this group is often denoted with

i

,

j

,

k

relations as follows:

Q

8

=

⟨

-

1

,

i

,

j

,

k

:

i

2

=

j

2

=

k

2

=

-

1

,

i

j

=

k

=

-

1

j

i

⟩

.

These relations are responsible for many useful results such as defining cross products for three-dimensional manipulations, and are also responsible for the
most common example of a

division ring

. As a group,

Q

8

is a curious specimen of a

p

-group in that it has only

normal subgroups

yet is

non-abelian

, it has a unique

minimal

subgroup

and cannot be represented

faithfully

except by a

regular representation

– thus requiring degree 8. [To see this note that the unique minmal subgroup is necessarily normal, thus if a

proper subgroup

is the

stabilizer

of an action, then the minimal normal subgroup is in the kernel so the

representation

is not faithful.]

A common work around is to use

2

×

2

matrices over

ℂ

but to treat these as matrices over

ℝ

.

-

1

=

[

-

1

0

0

-

1

]

,

i

=

[

i

0

0

-

i

]

,

j

=

[

0

i

i

0

]

,

k

=

[

0

-

1

1

0

]

.

A worthwhile additional example is

n

=

3

. For this
produces a

group order

12 which is often overlooked.

2

Subgroup structure

Proposition 2

.

Q

4

⁢

n

is

Hamiltonian

– meaning all a

non-abelian group

whose subgroups are normal – if and only
if

n

=

2

.

Proof.

As

Q

4

⁢

n

/

Z

⁢

(

Q

4

⁢

n

)

≅

D

2

⁢

n

, then if

Q

4

⁢

n

is Hamiltonian then we require

D

2

⁢

n

to be as well. However when

n

>

2

we know

D

2

⁢

n

has non-normal subgroups,
for example

⟨

a

⁢

b

⟩

. So we require

n

≤

2

. If

n

=

1

then

Q

4

⁢

n

is cyclic and so trivially Hamiltonian. When

n

=

2

we have the usual quaternion
group of order 8 which is Hamiltonian by direct inspection: the

conjugacy classes

are

{

1

}

,

{

a

2

}

,

{

a

,

a

3

}

,

{

b

,

a

2

⁢

b

}

and

{

a

⁢

b

,

a

3

⁢

b

}

, more commonly
described by

{

1

}

,

{

-

1

}

,

{

i

,

-

i

}

,

{

j

,

-

j

}

and

{

k

,

-

k

}

. In any case,
all subgroups are normal.
∎

By way of

converse

it can be shown that the only finite

Hamiltonian groups

are

A

⊕

Q

8

where

A

is abelian without an element of order 4.
One sees already in

ℤ

4

⊕

Q

8

that the subgroup

⟨

(

1

,

i

)

⟩

is conjugate to the distinct subgroup

⟨

(

1

,

-

i

)

⟩

and so such groups are not Hamiltonian.

Proposition 3

.

1.

|

a

i

|

=

2

⁢

n

/

i

for

1

<

i

≤

2

⁢

n

and

|

a

i

⁢

b

|

=

4

for all

i

.

2.

Every subgroup of

Q

4

⁢

n

is either cyclic or a generalized
quaternion.

3.

The normal subgroups of

Q

4

⁢

n

are either subgroups of

⟨

a

⟩

or

n

=

2

i

and it is

maximal subgroups

(of index 2) of
which there are 2 acyclic ones.

Proof.

The order of elements of

⟨

a

⟩

follows from standard

cyclic group

theory.
Now for

a

i

⁢

b

we simply compute:

(

a

i

⁢

b

)

2

=

a

i

⁢

b

⁢

a

i

⁢

b

=

a

i

⁢

a

-

i

⁢

b

2

=

b

2

.
So

|

a

i

⁢

b

|

=

4

.

Now let

H

be a subgroup of

Q

4

⁢

n

. If

Z

⁢

(

Q

4

⁢

n

)

≤

H

then

H

/

Z

⁢

(

Q

4

⁢

n

)

is
a subgroup of

D

2

⁢

n

. We know the subgroups of

D

2

⁢

n

are either cyclic or
dihedral. If

H

/

Z

⁢

(

Q

4

⁢

n

)

is cyclic then

H

is cyclic (indeed it is a subgroup
of

⟨

a

⟩

or

H

=

⟨

a

i

⁢

b

⟩

). So assume that

H

/

Z

⁢

(

Q

4

⁢

n

)

is
dihedral. Then we have a dihedral presentation

⟨

x

,

y

:

x

m

=

1

,

y

2

=

1

,

y

-

1

⁢

x

⁢

y

=

x

-

1

⟩

for

H

/

Z

⁢

(

Q

4

⁢

n

)

. Now

pullback

this presentation to

H

and we find

H

is quaternion.

Finally, if

H

does not contain

Z

⁢

(

Q

4

⁢

n

)

then

H

does not contain an element
of the form

a

i

⁢

b

, so

H

≤

⟨

a

⟩

and so it is cyclic.

For the normal subgroup

structure

, from the relation

b

-

1

⁢

a

⁢

b

=

a

-

1

we
see

⟨

a

⟩

is normal. Thus all subgroups of

⟨

a

⟩

are normal as

⟨

a

⟩

is a normal cyclic subgroup. Next suppose

H

is a normal subgroup not

contained

in

⟨

a

⟩

. Then

H

contains
some

a

i

⁢

b

, and so

H

contains

Z

⁢

(

Q

4

⁢

n

)

. Thus

H

/

Z

⁢

(

Q

4

⁢

n

)

is a normal
subgroup of

D

2

⁢

n

. We know this forces

H

/

Z

⁢

(

Q

4

⁢

n

)

to be contained in

⟨

a

⟩

/

Z

⁢

(

Q

4

⁢

n

)

, a

contradiction

on our

assumptions

on

H

, or

n

=

2

i

and

H

/

Z

⁢

(

Q

4

⁢

n

)

is a maximal subgroup (of index 2).
∎

Proposition 4

.

Q

4

⁢

n

has a unique minimal subgroup if and only if

n

=

2

i

.

Proof.

If

p

|

n

and

p

>

2

then

a

2

⁢

n

/

p

has order

p

and so the subgroup

⟨

a

2

⁢

n

/

p

⟩

is of order

p

, so it is minimal. As the center
is also a minimal subgroup of order 2, then we do not have a unique minimal subgroup
in these conditions. Thus

n

=

2

i

.

Now suppose

n

=

2

i

then

Q

4

⁢

n

is a

2

-group so the minimal subgroups must all
be of order 2. So we locate the elements of order 2. We have shown

|

a

i

⁢

b

|

=

4

for
any

i

, and furthermore that

(

a

i

⁢

b

)

2

=

b

2

=

a

n

. The only other minimal subgroups
will be generated by

a

i

for some

i

, and as

|

a

|

=

2

i

+

1

there is a unique minimal
subgroup.
∎

It can also be shown that any

finite group

with a unique minimal subgroup is either
cyclic of prime power order, or

Q

4

⁢

n

for some

n

=

2

i

.
We note that these groups have only

regular

faithful representations

.

Title

generalized quaternion group

Canonical name

GeneralizedQuaternionGroup

Date of creation

2013-03-22 16:27:41

Last modified on

2013-03-22 16:27:41

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

7

Author

Algeboy (12884)

Entry type

Derivation

Classification

msc 20A99

Synonym

quaternion groups

Related topic

DihedralGroupProperties

Defines

generalized quaternion

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