Let G be a finite group, i.e., |G| < ∞. 1). of the subgroup (a²) of G in G Let |G| = 360. Let a E G and |a| = 60. Find the number of left cosets 2). of G, i.e., K

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Let G be a finite group, i.e., |G| < o∞.
(1). Let |G| = 360. Let a e G and |a| = 60. Find the number of left cosets
of the subgroup (a²) of G in G
(2). Let K be a proper subgroup of H and let H be a proper subgroup
of G, i.e., K <H and H < G. It is known
(a). |K| = 30,
(b). |G| = 300, and
(c). there exists an element a E H with |a| = 25
%3D
Determine the possible order(s)of H, i.e., |H|.
Transcribed Image Text:Let G be a finite group, i.e., |G| < o∞. (1). Let |G| = 360. Let a e G and |a| = 60. Find the number of left cosets of the subgroup (a²) of G in G (2). Let K be a proper subgroup of H and let H be a proper subgroup of G, i.e., K <H and H < G. It is known (a). |K| = 30, (b). |G| = 300, and (c). there exists an element a E H with |a| = 25 %3D Determine the possible order(s)of H, i.e., |H|.
Expert Solution
Step 1 Given

A finite group i.e.,  G<

To Find- the no. of cosets of the subgroup a12 of G in G

           - the possible order(s) of H , i.e.,H. for K , a proper subgroup of H be a proper subgroup of G i,e, K < H and  H< G. it is known  (a)K=30

                  (b)G=300

                  (c) there exists an element aH with a=25

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