Re: Puzzled (4^162 mod 100)

In article <49baidFntrgfU1@news.dfncis.de>,
Sebastian Gottschalk <seppi@seppig.de> wrote:
>Paul Rubin wrote:
>> Sebastian Gottschalk <seppi@seppig.de> writes:
>>> The size of the cyclic subgroup of GF(100) generated by the element
>>> 4 is actually 20 which divides both 40 and 100. Taking the 40
>>> instead of 20 is just an elegant shortcut to see how blatant this
>>> example is. 4^142 mod 100 would have been at least a little bit
>>> trickier. :-)
>>
>> Hmmm, Z/100Z is not a group under multiplication; for example, 4 has
>> no inverse. (Terminology: GF(100) would mean a field with 100
>> elements, and no such field exists.)
>
>Hoops, right, that was Z_100.
>
>> I.e. that pattern cycles with length 10, not 20.
>
>Typo. I actually wondered a bit...
>
>> The multiplicative identity 1 doesn't occur in the list of powers, so
>> it's not a subgroup.
>
>The multiplicative identity of this subgroup is 76.

It's not a "subgroup" because there is no overgroup. It's a
subsemigroup which happens to be a group, but is not a submonoid since
the identity is not the identity of the monoid in which it is
embedded).

-- 
======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu