Any small multiplicative subgroup is not a sumset Abstract   We prove that for an arbitrary    e  u003e  0    and any multiplicative subgroup    xcex93  xe2x8ax86    F    p     ,    1  xe2x89xaa    xcex93    xe2xa9xbd    p    2    3  xe2x88x92  e      there are no sets B,    C  xe2x8ax86    F    p      with      B    ,    C    u003e  1    such that    xcex93    B    C   . Also, we obtain that for    1  xe2x89xaa    xcex93    xe2xa9xbd    p    6    7  xe2x88x92  e      and any    xcexbe  xe2x89xa0  0    there is no a set B such that    xcexbe  xcex93    1    B    B   .

New necessary conditions on (negative) Latin square type partial difference sets in abelian groups Abstract   Partial difference sets (for short, PDSs) with parameters (     n    2     ,    r  (  n  xe2x88x92  xcfxb5  )   ,    xcfxb5  n      r    2    xe2x88x92  3  xcfxb5  r   ,      r    2    xe2x88x92  xcfxb5  r   ) are called Latin square type (respectively negative Latin square type) PDSs if    xcfxb5    1    (respectively    xcfxb5    xe2x88x92  1   ). In this paper, we will give restrictions on the parameter r of a (negative) Latin square type partial difference set in an abelian group of nonprime power order      a    2      b    2     , where    gcd  xe2x81xa1  (  a  ,  b  )    1   ,    a  u003e  1   , and b is an odd positive integer xe2x89xa53. Very few general restrictions on r were previously known. Our restrictions are particularly useful when a is much larger than b. As an application, we show that if there exists an abelian negative Latin square type PDS with parameter set    (  9    p    4  s    ,  r  (  3    p    2  s      1  )  ,  xe2x88x92  3    p    2  s        r    2      3  r  ,    r    2      r  )   ,    1  xe2x89xa4  r  xe2x89xa4    3    p    2  s    xe2x88x92  1    2     ,    p  xe2x89xa1  1    (   mod     4  )    a prime number and s is an odd positive integer, then there are at most three possible values for r. For two of these three r values, J. Polhill gave constructions in 2009  10 .