mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in the field of

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

is said to be strictly simple if it has no proper nontrivial

ascendant subgroup In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its successor ...

s. That is,

G

is a strictly simple group if the only ascendant subgroups of

G

are

\

(the trivial subgroup), and

G

itself (the whole group). In the finite case, a group is strictly simple if and only if it is

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

. However, in the infinite case, strictly simple is a stronger property than simple.

References

Simple Group
Encyclopedia of Mathematics, retrieved 1 January 2012 Properties of groups {{group-theory-stub

See also

References