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Is there an algorithm to calculate the inverse of an element in the group algebra?

For example, does the element $(1 2 3) + 2 . (1 2)(3 4)$ in the group algebra $\mathbb{C} S_{4}$ have an inverse, and if so, how do I calculate it?

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    Where (in which group) is all this happening? Anyway look at my answer and leave your comments as to how satisfied you're! I'll edit to add relevant information whenever needed. Explain your notation as well!2012-01-23
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    I'm after inverses in the group _algebra_, not in the group. In the example, the underlying group is S4, and the field might as well be C. So the group algebra is CS4, consisting of C-linear combinations of elements of S4.2012-01-23
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    My answer is vanishing! I am good at misreading things! Good that you didn't downvote!2012-01-23

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