# Let α= (1234) and β= (24) ε S4, What is βα?

1) Let α= (1234) and β= (24) ε S4
a) Find |α| and |β|
b) Find H= < α , β>
c) Find |H|
2) Show that (Q, +) is not finitely generated.
3) Let G denote a group such that |G| < 200. Suppose G has subgroups of order 25 and...
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1) Let α= (1234) and β= (24) ε S4

a) Find |α| and |β|

b) Find H= < α , β>

c) Find |H|

2) Show that (Q, +) is not finitely generated.

3) Let G denote a group such that |G| < 200. Suppose G has subgroups of order 25 and 35. Find the order of G.

4) Give an example of a noncommutative group in which every

subgroup is normal

5) Let G denote a group. Let H < G be a subgroup of G such that H Z(G). Show that if G/H is cyclic then G=Z(G), i.e., G is commutative

a) Find |α| and |β|

b) Find H= < α , β>

c) Find |H|

2) Show that (Q, +) is not finitely generated.

3) Let G denote a group such that |G| < 200. Suppose G has subgroups of order 25 and 35. Find the order of G.

4) Give an example of a noncommutative group in which every

subgroup is normal

5) Let G denote a group. Let H < G be a subgroup of G such that H Z(G). Show that if G/H is cyclic then G=Z(G), i.e., G is commutative

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