# Abstract Algebra: Symmetric Group Problem

a) Let G = S_4. What orders do the elements have? Give reasons and examples.

b) Without listing them, how many subgroups does G have of order 3? Why?

c) Using examples and/or theorems, argue that G has at least one subgroup of every order dividing |G|.

© BrainMass Inc. brainmass.com October 10, 2019, 2:39 am ad1c9bdddfhttps://brainmass.com/math/group-theory/abstract-algebra-symmetric-group-problem-387632

#### Solution Preview

a) Since G = S_4, then each element can be expressed as multiplication of disjoint cycles. Elements of S_4 can

only have 2-cycle, 3-cycle and 4-cycle. So each element can be a 2-cycle, 3-cycle, 4-cycle ...

#### Solution Summary

Solution provides detailed answers for the questions listed.

$2.19