## The new encyclopaedia Britannica: in 32 vol. Macropaedia, Accounting - Architecture, Volume 13 |

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Page 275

A subset H of the elements of a group G that is itself a group under the

composition for G is called a

a group that have certain properties form a

one ...

A subset H of the elements of a group G that is itself a group under the

composition for G is called a

**subgroup**. In many important cases the elements ofa group that have certain properties form a

**subgroup**. For example, the one-to-one ...

Page 276

group G of order 12 to have 3

order 3. The mapping (h)a = xr^hx from the

is an isomorphism because products are preserved (see 178) and also inverses

are ...

group G of order 12 to have 3

**subgroups**of order 4 and also 4**subgroups**oforder 3. The mapping (h)a = xr^hx from the

**subgroup**H onto the**subgroup**x~lHxis an isomorphism because products are preserved (see 178) and also inverses

are ...

Page 277

For an equation of fifth degree G will in general be of order 120 and will have a

normal

so that in this natural way G, and G2 may be considered as

...

For an equation of fifth degree G will in general be of order 120 and will have a

normal

**subgroup**of order 60, a simple group ... x2) a**subgroup**isomorphic to G2,so that in this natural way G, and G2 may be considered as

**subgroups**of G, X G2,...

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