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## Homework Statement

Show that symmetry operations for en greek vase build up a symmetry group.

## Homework Equations

For en greek vase we have

[tex] \Gamma=[e, C_{2},\sigma, \sigma^{'}][/tex]

And there are 3 conditions which must be fullfilled so that the elements will create a symmetry group [tex]

1) (a\cdot b)\cdot c= a\cdot (b\cdot c)[/tex]

[tex] 2) a\cdot e= a[/tex]

[tex] 3) a\cdot a^{-1}=e[/tex]

## The Attempt at a Solution

So we know that the vase is invariant under [tex]180^{0}[/tex] so it is of [tex]C_{2} type[/tex]

do I understand correctly[tex]C_{2}\cdot C^{-1}_{2}=e[/tex] rotation [tex]180^{0}[/tex] and another one [tex]180^{0}[/tex] in the opposite direction

second condition-([tex]a\cdot e=a[/tex])

can we write then [tex]C_{2}\cdot e=C_{2}[/tex]???

How will it work for the condition 1?[tex](a\cdot b)\cdot c=a(b\cdot c)[/tex]

Can we show it in this way? [tex](C_{2}\cdot e)\cdot C^{-1}_{2}=C_{2}\cdot (e\cdot C^{-1}_{2})\rightarrow e=e[/tex]

How can we show it with using other symmetry elements? [tex][\sigma, \sigma{'}, e][/tex]

for example[tex] (C_{2}\cdot \sigma)\cdot e=C_{2}\cdot(\sigma\cdot e)[/tex]???