On $\kappa$-homogeneous, but not $\kappa$-transitive permutation groups

Abstract: A permutation group

G

on a set

A

is

k a p p a

-homogeneous iff for all

X, Y i n [A]^{k a p p a}

with

| A s e t m i n u s X | = | A s e t m i n u s Y | = | A |

there is a

g i n G

with

g [X] = Y

.

G

is

k a p p a

-transitive iff for any injective function

f

with

d o m (f) c u p r a n (f) i n [A]^{l e k a p p a}

and

| A s e t m i n u s d o m (f) | = | A s e t m i n u s r a n (f) | = | A |

there is a

g i n G

with

f s u b s e t g

. Giving a partial answer to a question of P. M. Neumann we show that there is an

o m e g a

-homogeneous but not

o m e g a

-transitive permutation group on a cardinal

l a m b d a

provided (i)

l a m b d a < {o m e g a}_{o m e g a}

, or (ii)

2^{o m e g a} < l a m b d a

, and

{m u}^{o m e g a} = {m u}^{+}

and

B o x_{m u}

hold for each

m u l e l a m b d a

with

o m e g a = c f (m u) < m u

, or (iii) our model was obtained by adding

{o m e g a}_{1}

many Cohen generic reals to some ground model. For

k a p p a > o m e g a

we give a method to construct large

k a p p a

-homogeneous, but not

k a p p a

-transitive permutation groups. Using this method we show that there exists

{k a p p a}^{+}

-homogeneous, but not

{k a p p a}^{+}

-transitive permutation groups on

{k a p p a}^{+ n}

for each infinite cardinal

k a p p a

and natural number

n g e 1

provided

V = L

.

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