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Is there a way to change the coordinates of a shapefile (points) from lat/long to UTM or Gauß-Krüger within the same CRS (i.e. WGS84:4326)?

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In general, you can reproject any shapefile to your desired CRS/Projection by just saving a new copy of it with that CRS applied (layer -> right click -> save as... and set desired CRS in dialog).

But what do you mean by 'within the same CRS'? UTM projection uses WGS84 or GRS80 as reference elipsoid, Gauss-Krüger however is projected and bound on/to either the Bessel or Krassowski elipsoid. You can reproject your lat/long to both, but for Gauss-Krüger you cannot (and don´t need to) keep the WGS84 reference system.

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    Just for completeness, you can have decimal degrees based on Bessel and Krasowski ellipsoid as well, choose EPSG:4314 (DHDN) or EPSG:4383 (Pulkovo) as target CRS. – AndreJ Aug 14 '17 at 10:13

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