Question

1. a noncyclic commutative group with more than 10 elements (a) List all elements in the group, their inverses, and find a smallest set of generators for your group. (b) Find a representation of your group isomorphic to a subgroup of $S_n$ for the smallest $n$ possible. It suffices to map each generator of your group to a suitable permutation in $S_n$ (given in cycle notation). (c) Find all subgroups of each group (it is not required to make a table of your group), then find a standard group that is isomorphic to each of these subgroups. (d) Make a diagram of the subgroups with smaller subgroups lower down on the page and a straight line from a subgroup $H$ to a subgroup $K$ if $H$ is a proper subset of $K$ and there is no other subgroup $L$ such that $H \subset L \subset K$.

          1. a noncyclic commutative group with more than 10 elements
(a) List all elements in the group, their inverses, and find a smallest set of generators for
your group.
(b) Find a representation of your group isomorphic to a subgroup of $S_n$ for the smallest $n$
possible. It suffices to map each generator of your group to a suitable permutation in $S_n$ (given in
cycle notation).
(c) Find all subgroups of each group (it is not required to make a table of your group), then find
a standard group that is isomorphic to each of these subgroups.
(d) Make a diagram of the subgroups with smaller subgroups lower down on the page and a
straight line from a subgroup $H$ to a subgroup $K$ if $H$ is a proper subset of $K$ and there is no
other subgroup $L$ such that $H \subset L \subset K$.

1. a noncyclic commutative group with more than 10 elements
(a) List all elements in the group, their inverses, and find a smallest set of generators for
your group.
(b) Find a representation of your group isomorphic to a subgroup of Sn for the smallest n
possible. It suffices to map each generator of your group to a suitable permutation in Sn (given in
cycle notation).
(c) Find all subgroups of each group (it is not required to make a table of your group), then find
a standard group that is isomorphic to each of these subgroups.
(d) Make a diagram of the subgroups with smaller subgroups lower down on the page and a
straight line from a subgroup H to a subgroup K if H is a proper subset of K and there is no
other subgroup L such that H ⊂ L ⊂ K.

Added by Sarah A.

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