A group G of order 20 is __________(a) solvable(b) unsolvable(c) 1(d) not determinedI had been asked this question during an online interview.My query is from Cyclic Groups in chapter Groups of Discrete Mathematics

A group G of order 20 is __________(a) solvable(b) unsolvable(c) 1(d)

1.	A group G of order 20 is __________(a) solvable(b) unsolvable(c) 1(d) not determinedI had been asked this question during an online interview.My query is from Cyclic Groups in chapter Groups of Discrete Mathematics
Answer» The correct option is (a) solvable The best I can EXPLAIN: The prime factorization of 20 is 20=2⋅5. Let n5 be the number of 5-Sylow subgroups of G. By Sylow’s theorem, we have,n5≡1(mod 5)and n5\|4. THUS, we have n5=1. Let P be the unique 5-Sylow subgroup of G. The subgroup P is normal in G as it is the unique 5-Sylow subgroup. Then consider the subnormal series G▹P▹{e}, where e is the identity element of G. Then the factor GROUPS G/P, P/{e} have order 4 and 5 respectively. HENCE these are cyclic groups(in particular ABELIAN). Hence, the group G of order 20 has a subnormal series whose factor groups are abelian groups, and thus G is a solvable group.

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