Lf groups, aec amalgamation, few automorphisms

We deal mainly with https://www.w3.org/1998/Math/MathML”> K λ l f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_6.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/>, the class of locally finite groups of cardinality https://www.w3.org/1998/Math/MathML”> λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_7.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/>, in particular https://www.w3.org/1998/Math/MathML”> K λ e x l f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_8.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/>, the class of existentially closed locally finite groups. In https://www.w3.org/1998/Math/MathML”> § https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_9.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> 3 we prove that for almost every cardinal https://www.w3.org/1998/Math/MathML”> λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_10.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> “every locally finite https://www.w3.org/1998/Math/MathML”> G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_11.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> of cardinality https://www.w3.org/1998/Math/MathML”> λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_12.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> can be extended to an existentially closed complete group of cardinality https://www.w3.org/1998/Math/MathML”> λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_13.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> which moreover is so called https://www.w3.org/1998/Math/MathML”> (λ, θ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_14.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> -full; note that https://www.w3.org/1998/Math/MathML”> § https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_15.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> 3 which do not rely on https://www.w3.org/1998/Math/MathML”> § https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_16.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> 1, https://www.w3.org/1998/Math/MathML”> § https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_17.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> 2. (in earlier results https://www.w3.org/1998/Math/MathML”> G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_18.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> has cardinality https://www.w3.org/1998/Math/MathML”> < λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_19.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> and also https://www.w3.org/1998/Math/MathML”> λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_20.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> was restricted). In https://www.w3.org/1998/Math/MathML”> § https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_21.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> 1 we deal with amalgamation bases, for the class of lf (= locally finite) groups, and general suitable classes, we define when it has the https://www.w3.org/1998/Math/MathML”> (λ, κ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_22.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> -amalgamation property which means that “many” models https://www.w3.org/1998/Math/MathML”> M K λ k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_23.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> are amalgamation bases and get more than expected. In this case, we deal with a general frame - so called a.e.c., abstract elementary class. In https://www.w3.org/1998/Math/MathML”> § https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_24.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> 2 we deal with weak definability of https://www.w3.org/1998/Math/MathML”> a N \ M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_25.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/> over https://www.w3.org/1998/Math/MathML”> M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429263637/26bd4099-d326-4c28-a33f-7985ba070843/content/matha0_26.tif” xmlns:xlink=“https://www.w3.org/1999/xlink”/>, for = existentially closed LF group.