Let G be a group of order lGl = 992 = 2^5 x 31. (a) Show that at least one of it

Question

Let G be a group of order lGl = 992 = 2^5 x 31. (a) Show that at least one of its Sylow p-subgroups is normal. (b) If G is Abelian, find all prime-power decompositions and invariant decompositions. (Show complete solutions.

Please make sure that it is completely different than this one.

Let G be a group of order 99. (a) prove that there exists a subgroup of H of order 3. (b) prove there is a unique subgroup K of G such that K/H = Z(3) and G/K = Z(11). The solutions provided are
identical when answered but are different problems. Thank you!!!

Explanation / Answer

Consider G, it has a 2-sylow subgroup and one 31-sylow subgrp We have a thm that states that the number of p-sylow subgrp is of the form pk+1 so lets consider the 31-sylow subgrp The number of these is of form 31k+1 (so its 1 or 32) [the only common element in these subgrps is e as the 31-sylow subgrp is a cyclic hence all the non identity elements have order 31] Case 1: If the number of 31 sylow subgrp is 1 then its a unique subgrp of order 31 We know that unique order subgrps are normal consider H be the only subgrp of order n. consider aH(a^-1) its also a subgrp of order n hence its aH(a^-1) = H hence H is normal. Hence if the sub grp is 1 then 31 - sylow subgrp is normal else lets prove that 2-sylow is normal. number of the 31-sylows be 32 they only share e. hence the number of elements in these are (31-1)*32+1=961 hence the 2 sylow grps : can only be unique as the order of the 960 elements is 31 hence they cannot be in 2-sylow(order = 32, the element order divides it) hence we have a unique 2-sylow which is normal hence proved message me if you have any doubts

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