If M is a proper subgroup of G we call it maximal if M≤X≤G implies X=M or X=G.

anon anon

anon anon

Answered question

2022-06-13

Answer & Explanation

Ian Adams

Ian Adams

Skilled2022-06-14Added 163 answers

Step 1

 

Solution: If G has no maximal subgroups, I(G) is taken to be G.⇒I(G) characterizes G in this case. Assume G has at least one maximal subgroup MLet α∈sub(G), Then α(M) is a maximal subgroup of G.If N≤G with α(M)≤N≤G, Then M≤α-1(N)≤G⇒α-1(N)=M or α-1(N)=G⇒N=α(M) or N=GSo, α(M) is maximal.
 

Step 2

 

1)α-1∈Sub(G), so α-1(M) is maximal.2)(α.α-1)(M)=(α-1.α)(M)So, α permutes the maximal subgroups of G.Let x∈I(G), then x∈∩M≤GMSo, α(x)∈α(∩M≤GM)=∩M≤Gα(M)=∩M≤GM=I(G)
 

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