Simple groups and JH theorem

Question 1

G is simple if

Answer 1

G=/={1} and the only normal subgroups of G are G and {1}

Question 2

commutator of n and g

Answer 2

[n,g] = n^-1 g^-1 n g

Question 3

if n greater or equal to 3 then

Answer 3

every element of An can be written as a product of 3-cycles

Question 4

if n greater of equal to 5 then

Answer 4

then the cycles are all conjugate in An

Question 5

A5 is …

Answer 5

simple

Question 6

for n greater or equal to 5, An is

Answer 6

simple

Question 7

G1,….,Gn subgroups of G is called a

Answer 7

composition series for G

Question 8

K is a maximal normal subgroup of G<=>

Answer 8

G/K being simple

K1/K is a normal subgroup of G/K
K ≤ K1 is a normal subgroup of G

Question 9

JH Theorem

Answer 9

G finite, =/= {1}
two composition series G |≥ H1 |≥……|≥Hr
and similar but K and s then
i) r =s
ii) G/H1, H1/H2 …. G/K1, K1/K2 are the same simple groups up to isomorphism

Question 10

K is a maximal normal subgroup of G <=>

Answer 10

G/K is simple

Question 11

N1 G, N2 G

Answer 11

N1N2 normal subgroup of G

Question 12

N1 G, N2 G

Answer 12

intersection of N1 N2 is a normal subgroup of G

Question 13

N G, H subgroup of G, then

Answer 13

HN/N isomorphic to H/H∩N