reserve G for Group,
  a,b for Element of G,
  m, n for Nat,
  p for Prime;

theorem Th10:
  for G being finite Group, N being normal Subgroup of G holds
  N is Subgroup of center G & G./.N is cyclic
  implies G is commutative
proof
  let G be finite Group;
  let N be normal Subgroup of G;
  assume that
A1: N is  Subgroup of center G and
A2: G./.N is cyclic;
  reconsider I = G./.N as finite Group;
  consider S be Element of I such that
A3: for T being Element of I ex n being Nat st T = S |^ n
   by A2,GR_CY_1:18;
  consider a1 be Element of G such that
A4: S = a1 * N & S = N * a1 by GROUP_6:21;
  for a,b being Element of G holds a * b = b * a
  proof
    let a,b be Element of G;
A5: a * N is Element of I & b * N is Element of I by GROUP_6:22;
    then consider r1 be Nat such that
A6: a * N = S |^ r1 by A3;
    a * N = (a1 |^ r1) * N by A4,Th8,A6;
    then consider c1 be Element of G such that
A7: a = (a1 |^ r1) * c1 & c1 in N by Th9;
A8: c1 in center G by A1,A7,GROUP_2:40;
    consider r2 be Nat such that
A9: b * N = S |^ r2 by A3,A5;
    b * N = (a1 |^ r2) * N by A4,Th8,A9;
    then consider c2 be Element of G such that
A10: b = (a1 |^ r2) * c2 & c2 in N by Th9;
A11: c2 in center G by A1,A10,GROUP_2:40;
    a * b = (a1 |^ r1) * (c1 * ((a1 |^ r2) * c2)) by A7,A10,GROUP_1:def 3
         .= (a1 |^ r1) * (((a1 |^ r2) * c2) * c1) by A8,GROUP_5:77
         .= (a1 |^ r1) * ((a1 |^ r2) * (c2 * c1)) by GROUP_1:def 3
         .= ((a1 |^ r1) * (a1 |^ r2)) * (c2 * c1) by GROUP_1:def 3
         .= (a1 |^ (r1 + r2)) * (c2 * c1) by GROUP_1:33
         .= ((a1 |^ r2) * (a1 |^ r1)) * (c2 * c1) by GROUP_1:33
         .= (a1 |^ r2) * ((a1 |^ r1) * (c2 * c1)) by GROUP_1:def 3
         .= (a1 |^ r2) * ((a1 |^ r1) * c2 * c1) by GROUP_1:def 3
         .= (a1 |^ r2) * (c2 * (a1 |^ r1) * c1) by A11,GROUP_5:77
         .= (a1 |^ r2) * (c2 * ((a1 |^ r1) * c1)) by GROUP_1:def 3
         .= b * a by A7,A10,GROUP_1:def 3;
    hence thesis;
  end;
  hence thesis by GROUP_1:def 12;
end;
